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Table of contents :

SEMICOND.pdf......Page 0

Electron Mobility......Page 6

Current Density......Page 7

Conductivity......Page 8

Resistance as a Function of Temperature......Page 9

Energy Bands and Bandgap Energy......Page 11

Electron-Hole Pair Generation and Recombination......Page 12

Photoconductors and Photodetectors......Page 14

Temperature Dependence of Hole-Electron Concentration......Page 17

Carrier Concentrations......Page 21

Fermi Level......Page 24

Non-Ideal Factors That Affect Terminal Characteristics......Page 33

SEMICOND.pdf......Page 0

Electron Mobility......Page 6

Current Density......Page 7

Conductivity......Page 8

Resistance as a Function of Temperature......Page 9

Energy Bands and Bandgap Energy......Page 11

Electron-Hole Pair Generation and Recombination......Page 12

Photoconductors and Photodetectors......Page 14

Temperature Dependence of Hole-Electron Concentration......Page 17

Carrier Concentrations......Page 21

Fermi Level......Page 24

Non-Ideal Factors That Affect Terminal Characteristics......Page 33

- Similar Topics
- Technique
- Electronics

INTRODUCTION Electronics is heavily relied on by most other areas of electrical engineering. While there is a considerable body of theory in communications, controls, etc., these areas ultimately use electronics to actually implement the functions. Electronic circuits use electronic devices to perform functions on signals such as amplification, filtering, rectifying, switching, etc.. Electronics has been a major topic of study in Electrical Engineering for nearly a century. Early electronic circuits used devices such as spark gaps and point-contact crystal diodes to perform signal processing. Later on, vacuum tubes were invented which made electronic communications and control systems widely available. In the late 1940’s, semiconductor devices such as diodes and transistors became available which created an electronics revolution. With these changes in technology, the study of electronics did not change significantly, only the devices changed. The circuits and the methods did not change substantially. The study of electronics can be roughly divided into two areas, devices and circuits. The study of devices is concerned with physical processes such as electron flow while the study of circuits emphasizes using the devices in applications and signal processing functions. The study of electronic circuits is further subdivided into analog, or linear, and digital, or switching, electronics. This course focuses on digital electronic circuits. By far, the greatest use of digital electronic circuits occurs in digital computers. Logic circuits are widely available from simple logic gates in small-scale integrated (SSI) circuits to very complex digital functions in very large-scale integrated (VLSI) circuits. In almost all digital circuits, transistors and diodes operate in two modes, on or off, carrying current or not carrying current; in essence, a switch. We will look at how digital logic circuits operate and what the terminal characteristics and manufacturer’s specifications mean. We will then look at how to go beyond the logic circuits with interfaces both at the inputs and outputs. We begin this course with a brief discussion of semiconductor materials and pn junctions. This material is neither rigorously developed nor complete. A rigorous study of semiconductor electronics is left for later. However, to effectively use semiconductor devices, it is necessary to have a basic understanding of how they work. Because electronic devices are non-linear, we will look at their terminal characteristics and make circuit models of the devices that will allow us to use to linear circuit analysis techniques to analyze the circuits. We will then look at application of semiconductor devices in switching circuits including logic gates, interface circuits, and special applications. Many electronic systems involve both analog and digital circuits and during this course, we will look at some of these cases requiring a mixture of applications. The ultimate test of understanding the material of this course will be found in the design exercises.

Introduction

1

Chapter 1

Semiconductors THE ELECTRON IN ELECTRIC FIELDS If we were to take two parallel plates and connect a voltage source across them as shown in Figure 1, an electric field would be set up between the plates. Neglecting fringing around the edges, the electric field would be uniform everywhere between the plates. The electric field strength would be E = V/d

(1)

where V is the applied voltage and d is the distance between the plates. Thus, the electric field strength E has the units volts per centimeter and is a vector quantity going from a positive charge to a negative charge. Note: CGS units are normally used in semiconductor physics - centimeters, grams, seconds.

Now if a tiny person, let's call her Millie Micron, was able to carry an electron into the region between the plates and release it as shown in Figure 2, the electron would be attracted to the positive plate and repelled by the negative plate. The force on the electron would be Fx = -qEx

(2)

where q is the electronic charge. The negative sign occurs because the electron is accelerated in the negative x direction, toward the positive plate. Of course, the electron would obey Newton's laws and the acceleration, ax, would be a function of the mass of the electron, m, and the force exerted by the electric field, Fx = -max

(3)

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Chapter 1

Figure 2. Millie releasing an electron within the electric field As the electron accelerates, it gains kinetic energy. Just as with objects with mass in a gravitational field, the electron in an electric field has potential energy that can be converted to kinetic energy. The total energy then is W = U + 1/2mv2

(4)

where U is the potential energy and v is the velocity. The energy associated with a single electron is quite small compared to units we normally work with so we use the units electron volts defined as moving one electron across a potential difference of one volt. 1 eV = 1.602 x 10-19 joules (1 joule = 1 watt second) The electronic charge is 1.602 x 10-19 coulombs (ampere-seconds). In our example, if the voltage source is 5 volts and Millie released the electron at the negative plate, the electron would gain five electron volts of energy as it fell to the positive plate. At that point it would have zero potential energy. Thus, at the point of release, the electron had a potential energy of 5 eV. This 5 eV would be converted to kinetic energy by the time it arrived at the positive plate. To look at this another way, let's look at a plot of the electric potential within the field. We will assume the positive plate is grounded and at zero potential. The negative plate is at negative five volts with the potential changing linearly in between as shown in Figure 3. In this example, let's assume Millie is standing on the positive plate and throws the electron toward the negative plate. If she throws it gently, it will start with only a small kinetic energy which is soon converted to potential energy as the electron goes against the electric field. When all the kinetic energy is converted to potential energy, the electron has zero velocity. The electric field accelerates the electron back toward the positive plate. The effect is that the electron falls back to Millie and she catches it. If she then throws it again, but this time a little harder, it will go further, but will again fall back. Say, this is a neat game isn't it?. This is similar to throwing a ball up a sloping roof and

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Chapter 1 catching it as it rolls back down. Just as with the roof, if Millie throws the electron hard enough, it will overcome the potential hill and escape. In this case, we must envision the negative plate as having lots of holes, like a wire screen, the electron can go through to escape; for example. This potential barrier concept will be used when we look at p-n junctions.

Figure 3. Illustration of the potential energy barrier. ELECTRON EMISSION FROM THE METAL Now, we can describe current flow between two metal plates. If a voltage is applied across the two metal conductors with just vacuum in between the two conductors, obviously no current flows between the two plates in the vacuum. There will be an electric field (E = Vs/x, where x is the distance between the two plates) dropped across two conductors. Charge builds up on each surface of the metal conductors that are facing each other. On one conductor, the cathode, electrons collect on the surface. On the other conductor, the anode, electrons are repelled from the surface, leaving the fixed metal ions at the surface. If a strong enough voltage (or a sufficiently high electric field) is applied, the electrons have a total energy equal to the vacuum energy, Evac. The electrons at the surface of one metal conductor will be ripped out of the conductor and then move to the anode. And, current does flow across the vacuum! This is called thermionic emission, a process by which tungsten filaments emit beams of electrons in cathode ray tubes for television, oscilloscope screens, and other instruments. The electric field required to rip the electrons out of the metal and into the vacuum is equal to qφm, where φm is the work function of the metal and q is the charge on an electron. Different metals have different work functions. So, some metals work better than others as “electron guns”. When additional energy is added to the system, for example by heating the metal, the kinetic energy of the electrons in the metal is increased and the electric field required for

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Chapter 1 thermionic emission is decreased. Thus, there are some “electron guns” that are called cold cathode emitters and others called hot filament emitters, denoting the temperature of the metal from which the electrons are escaping. The tungsten filament in your television is a hot filament emitter, running at over 2500K. Cold cathode emitters are used when a high temperature filament is impractical in the system. The cold cathode emitters are being researched for thin film displays and other applications where the system can not handle a large thermal gradients. CURRENT CONDUCTION IN METALS Metal atoms have one or more very loosely bound valence electrons. These are the electrons in the outer most orbital or electron shell, in an s or d orbital. At normal temperatures, the valence electrons have enough thermal energy to be easily separated from the metal atom and move randomly throughout the material. The metal atom, then, becomes a positively charged ion. Figure 4 is a two-dimensional representation of the situation where the electrons are free and the metal ions are immobile. The material still has zero net charge as there are still as many electrons in the metal as there are positive charges on the metal ions. In its random motion, an electron occasionally collides with a metal ion, but with its thermal energy, it is not captured and rebounds at a random angle. The motion of electrons in metal described here is similar to the motion of molecules in gas. Thus, it is called the electron gas model.

metal ion - free electron

If we were to average the motion or velocity of the electrons in the metal in Figure 4, we would find zero net motion and zero average velocity. However, if we were to apply a voltage between the ends of the metal conductor, a field would be set up between the ends of the conductor and the electrons would be accelerated toward the positive end. Thus, there is a drift of electrons in the conductor toward the positive end and a current results. This current is called drift current. The average speed at which the electrons drift is called the drift velocity, vd. It seems as though the electrons might continue to accelerate in the field and reach very high velocities, but instead, the electrons soon collide with a fixed ion and recoil in a random direction, to again be accelerated by the field. Therefore, the electron's drift velocity reaches a maximum at some electric field and does not increase any further with increasing electric field. This is called the saturation velocity. The drift velocity under low field conditions, when the drift velocity is well below the

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Chapter 1 saturation velocity, is a function of the electric field and the physical properties of the conductor, vd = µE

(5)

where E is the electric field strength and µ is the mobility of the electron in that material. Electron Mobility Mobility is a physical constant that describes the ease in which an electron can move through a material. The mobility is a function of temperature as well as the electric field. Thus, the speed of the electrons in the metal changes with temperature. The exact relationship between the change in mobility as a function of temperature is dependent on a number of material properties including the number of grain boundaries and how pure the metal is. At 0K, there is no kinetic energy. Therefore, the valence electrons are localized in a metal atom. As the temperature is increased from 0K, there are some free electrons. However, their thermal energy is low. They interact with the metal ions and undergo Coulombic scattering events. In coulombic scattering, they lose some of their energy to the metal ion so their mobility is limited by the number of scattering events. A macroscopic example of a similar problem would be if you shot an iron marble (the electron) through a grid of magnets (the metal ions), where there is considerable space between each of the magnets. The magnets are regularly spaced and fixed in position. When you shoot the marble, your aim is great – the path is clear from one end of the grid to the other. However, if the marble is rolled slowly (low kinetic energy), it is likely that its motion will be perturbed by the magnetic field of the magnets. At some point in its travel, the magnetic attraction may be strong enough to divert the forward motion of the marble and the marble will collide with the magnet. As you increase the speed at which you send the marble through the grid (increase the kinetic energy/temperature), the farther the marble will go through the grid before it gets attracted to one of the magnets. As the temperature is increased further, the kinetic energy of the free electrons increases and the scattering events do not result in significant energy transfer. So, the mobility of the electron increases. However, as temperature increases significantly (to ~100K), the kinetic energy of the metal ions becomes large enough that they are vibrating with sufficient movement to impede the movement of the electrons. The scattering events are called lattice scattering. Again, there is energy transfer and the free electrons’ mobility is decreased because of the scattering events. Unlike Coulombic scattering, the effect of

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Chapter 1 lattice scattering increases as the temperature increases, further decreasing the mobility of the free electrons. Continuing with the macroscopic example: Now modify the grid so that the magnets are mounted on springs. As the grid is moved (because both the electrons and the metal ions have kinetic energy), the magnets “jiggle” back and forth. This reduces the amount of open area between each magnet. As the movement of the grid increases (an increase in temperature), the magnets jiggle further into the open areas and jiggle faster back and forth. This increases the likelihood that the marble will hit one or more of the magnets during its travel, reducing the distance that the marble can move through the grid. Current Density

Figure 5 shows a section of a conductor of length L and cross section A. Within this volume, there are N electrons. The Figure 5. A section of a number of free electrons, N, depends on the metal. If we conductor with current I apply a voltage, V, from end-to-end, the electrons will drift towards the positive end with a drift velocity of vd. The time required for the electronics to traverse the length, L, at an average drift velocity, vd is (6)

tt = L vd

All N of the electrons pass through the end of the section in time, tt. This electron drift constitutes a current I in amperes (coulombs/second).

(7) Substituting the transit time, tt, into equation (7)

(8) where J is the current density which is defined as current over the cross sectional area. The electron density is the number of electrons in a unit volume, n = N/LA. Thus

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Chapter 1

J = qnvd

(9)

Conductivity We can write current density in terms of the electric field by substituting vd = µE, J = qnµE

(10)

We know that voltage is the electric field, E, times the distance over which the electric field is dropped (V = EL). Thus, resistance, R, can be rewritten as follows:

(11) In equation 11, we have related resistance of a sample to its shape and a parameter called resistivity, ρ. We define this parameter, ρ, as well as a second parameter, conductivity (σ), which is the inverse of resistivity, as: σ = 1/ρ = qnµ

(12)

This result is an important equation. Conductivity is a function of the electronic charge, the (free) electron density, and the mobility of the electrons in the material. In metals, the free electron density is equal to the number of valence electrons times the density of atoms in the material that readily give up the valence electrons. Thus, mobility can be readily determined once the conductivity is found, which in turn is determined from the resistance measurement of a sample of the material. Also the physical dimensions of the metal are critical in determining its resistance. Resistance is a linearly proportional to the length of the metal and it is inversely proportional to its cross-sectional area. Example: Aluminum has a density of 2.7 g/cm3, atomic weight of 27, and a resistivity of 3.44x10-6 Ω−cm. If Aluminum has three valence electrons, what is the mobility? Avogadro’s number is 6.02x1023 atom per mole. Thus, one mole of aluminum weighs 27 grams and has 6.02x1023 atoms. From this we get that one cm3 of aluminum contains 0.1 moles or 6.02x1022 atoms. If each atom has three free electrons, there are 18.06x1022 free electrons per cubic meter. Thus we have σ = qnµ or

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Chapter 1 σ = (3.44x10-6 Ω cm)-1 = 1.602x10-19 Coul x 18.06x1022 free electrons/ cm3 x µ µ =10 cm2/Vs So, if we have 3 meters of Aluminum wire, the length of a normal oscilloscope cable, and there is 5 volts dropped across the cable, the time that it will take an electron at one end of the cable to make it all the way to the other end of the cable (called transit time, tt) is 1800 seconds. vd = µE tt = L/(µ*E)

(13)

tt = 300cm/(10cm2/Vs*(5V/300cm)) = 1800 s Not too speedy is it? This is one of several reasons why we don't transfer data using individual electrons, except over very short distances. Resistance as a Function of Temperature Now, what happens to the resistance of a metal if we increase its temperature from room temperature (roughly 25oC) to, say, 125 oC, a commonly used maximum temperature of operation for certain devices and circuits? Look back at equation 11. Based upon our previous discussions, you know that the number of free electrons in the metal do not increase with increasing temperature. Of course, the length and cross-sectional area do increase slightly as the metal expands. For example, the length of an aluminum bar increases by 12 parts per million per degree C. Thus, a 1 meter long aluminum bar at room temperature will increase to a length of 1.012 meters. However, the parameter that has most significant change with temperature is the mobility of the free electrons, µ. And, the mobility decreases as the temperature increases to 125oC. Thus, the resistance of the metal increases with increasing temperature. If you look at the specifications for metal- and carbon-based resistors, you will see that their resistance is not a constant as a function of temperature, but increases. This increase in resistance with temperature can be minimized over specific temperature ranges through a careful design of the materials used and the physical structure of the resistor. So, even the selection of resistors needs to considered when you design of circuit – the selection will depend on your application. INTRINSIC SEMICONDUCTORS Atoms in metals have relatively free valence electrons, which provide the carriers for conduction. Insulators are materials where there are virtually no free electrons, hence, no

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Chapter 1 conduction. Semiconductors are in between. The most common semiconductor material is silicon, with germanium, gallium arsenide and indium phosphide used for some special purposes. As you can see from the periodic table, both silicon and germanium are group IV materials and have four valence electrons, two in the outer most s shell and two electrons in the p orbital. Gallium arsenide is a III-V compound where gallium has three valence electrons (two in the s orbital and one in the p orbital) and arsenic has five (two in the s orbital and three in the p orbital). A maximum of two electrons occupy the s orbital and a maximum of six occupy the p orbital. The number of valence electrons is important because these materials form crystalline structures and each atom shares electrons with its neighbors in covalent bonds to, in effect, fill each atom's outer shell. Because silicon is by far the most widely used semiconductor material, we will use it in our examples. A two-dimensional representation of the of the silicon structure is shown in Figure 6. Here the silicon atoms are arranged in regular rows and columns as in a crystal. Each atom shares four electrons with its four nearest neighbors. In the actual three-dimensional crystal, the structure is a tetrahedron and is called a diamond lattice. There are eight silicon atoms in each unit cell of the diamond lattice. A unit cell is the smallest volume that you can make that still has the same geometry of the overall crystal. In Si, the unit cell is a cube with dimensions of a = 0.545nm per side, where a is called a lattice constant.

As shown in the drawing, each electron is part of a covalent bond and is rather tightly bound. However, due to thermal agitation, an occasional electron has enough energy to break free from the bond and become free to roam throughout the crystal as shown in Figure 7. The only place with a positive charge that can capture the free electron is another broken bond. Thus, these free electrons can contribute to electrical conduction.

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Chapter 1

Energy Bands and Bandgap Energy

The free electrons are called conduction electrons. The conduction electrons reside in the conduction band, at an energy of EC and above. The remaining electrons that are bound to the atoms, the valance electrons, reside in the valence band, at energies of EV and below. The amount of energy required to free the electron is called the bandgap energy, Eg. The bandgap energy arises from the interaction of the atoms within the solid, the covalent bonds that results in the sharing of electrons between neighboring atoms. The energy required for thermionic emission of electrons from the semiconductor crystal, the emission of an electron from a semiconductor material into vacuum, is equal to qχe, the charge on an electron times the electron affinity, χe. The graph to the right is called an energy diagram and shows these different energy levels. The difference between semiconductors and non-conductors, or insulators, is the size of the energy gap between the valence bands, and the conductions bands. If the energy gap is very small, then the material will conduct electric current easily. If the band gap is large, then it will be a non-conductor. Semiconductors typically have a band gap of about 1 electon volt, while insulators have a band gap of several electron volts. This definition is somewhat vague and should only be used as a general guideline. HOLE, THE MISSING ELECTRON One interesting feature of the semiconductor is that the broken bond, called a hole, can also contribute to conduction. These holes reside at the top of the valence band, with an energy of EV and are positively charged. Because the valence electrons and atoms have kinetic energy and the atoms are closely packed in a crystal, a valence electron can break its bond with a neighboring atom and complete the broken bond. Since there is now another broken bond on a different atom, a hole is now located on the neighboring atom. The results is that the hole has moved.

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Chapter 1 Just as the electron can be caused to drift in an electric field, so can the hole, although in the opposite direction because it has a positive charge. Thus, if an electric field is applied, the electrons move one direction and the holes the other. The result is that both contribute to current flow; the conventional current being in the direction of hole flow and opposite the direction of electron flow. This is unlike current flow in metals where only the free electrons contribute to current. The metal ions are fixed. Electron-Hole Pair Generation and Recombination At any time a hole and a free electron may come together and recombine, thus removing the conducting elements. In fact, recombination occurs naturally and is an on-going process where hole-electron pairs are continuously created and annihilated. When the hole and free electron recombine, the free electron needs to lose the excess energy that enabled it to break free of its silicon atom. This amount of energy is equal or greater than the bandgap of the semiconductor (Eg), the difference between EC and EV We use two quantum mechanical particles to describe how this energy is released. The energy may be released as phonons, a quantum of thermal energy or heat, or it may be released as photons, a quantum of light. Phonons are usually released when the semiconductor is an indirect semiconductor. Two examples of an indirect semiconductor are silicon and germanium. In these semiconductors, the location of the minimum energy point in the conduction band is not directly above the maximum energy point in the valence band, when plotting the energy of the conduction band and valence band as a function of momentum, k. Since momentum has to be conserved when the electron recombines with the hole, the difference in momentum between the electron and hole is given to another particle along with the excess energy. As a phonon has momentum and a photon does not, a phonon must also be emitted to conserve both energy and momentum. Photons are usually released when the semiconductor is a direct semiconductor. An example of a direct semiconductor is gallium arsenide. In these semiconductors, the location of the minimum energy point in the conduction band is directly above the maximum energy point in the valence band, when the conduction band and valence band energies are plotted as a function of momentum, k. The only particle that can be released when the electron and hole recombine that has energy, but no momentum

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Chapter 1 is a photon. Therefore, light is emitted by direct semiconductors, while heat is emitted by indirect semiconductors, during electron-hole pair recombination Thus, semiconductor laser diodes are composed of direct semiconductors such as gallium arsenide. As mentioned, the generation of electrons and holes requires energy and when an conduction electron is created so to is a hole created. This energy can be thermal energy, heat, or the energy can come in the form of light, where the energy of the photon has to be equal to or greater than the bandgap of the semiconductor, Eg. This process is more apparent in some applications such as photo-generated current in photoconductors. CARRIER LIFETIMES In any semiconductor, hole-electron pairs are in a state of constant change. The total number of electrons and holes is a constant. However, individual electrons are sometimes found in the conduction band and other times are in the valence band. If a hole and an electron should meet, they will recombine - reducing the number of carriers. At the same time, the thermal energy of the crystal is exciting additional electrons to break the covalent bonds. At equilibrium, the number of pairs created is equal to the number that recombine. Thus, any free electron or any free hole will eventually recombine. Carrier lifetime, τ, is a measure of how long any individual electron or hole will remain free. How would we measure this lifetime? Let us do a thought experiment. Assume we have a block of silicon at room temperature. Let’s attach a voltage source across it and measure the current as shown in Figure A below.

Figure A: A block of silicon with a voltage source Now we can determine the resistance of the silicon block. From the resistance, we can calculate the electron and hole densities. Then we can determine the total current contribution by the electrons and the holes. Now let’s use a powerful strobe lamp and create a flash of light with an energy greater than or equal to the bandgap of the semiconductor, Eg. When this light falls on the silicon, the photons excite the electrons in the covalent bonds. Some of these electrons

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Chapter 1 will be excited into the conduction band and create hole-electron pairs in excess of the steady-state intrinsic level. These excess carriers are available for conduction so the current through the silicon block increases as the resistance of the silicon decreases. However, over time, the number of electron-hole pairs decreases because, the system will return to equilibrium without the continued light exposure (since it was only a pulse) providing the extra energy to free the electrons and create the holes. The excess electrons and holes will recombine and the electron and hole concentration returns to the equilibrium carrier concentration, ni. Thus, the current will decrease in an exponential decay as shown in Figure B. We can measure carrier lifetime by measuring the rate at which the current decays. Carrier lifetime is one time-constant of the current pulse decay. (τ = 1/α) Note the similarity to the time constants for capacitive and inductive circuits.

Figure B. Current following a pulse of light. This experiment also leads us to the idea of photoconductors and photodetectors. Photoconductors and Photodetectors In a photoconductor, a voltage is dropped across the length of semiconductor. When a photon of light hits a covalent bond, it can transfer its energy to an electron by absorption, which can cause the covalent bond to disassociate and create a hole-electron pair. Because of the electric field across the semiconductor, the electron and hole move - in opposite directions. For as long as the photo-generated electron and hole exist, a current will flow in the photoconductor. When the electron and hole recombine, the current caused by the photogenerated carriers will cease. Of course, recombination occurs randomly recombining any hole with any electron, not necessarily the original pair. First of all, they are not monogamous. Secondly, they aren't even close to each other after a few femtoseconds, as they drift in opposite directions. In order for an electron to recombine with a hole to create a covalent bond between two Si atoms, they have to be within an atomic length of each other. In the case of Si, the atomic length is 0.235nm. As mentioned above, the average length of time that these photogenerated carriers will exist before they recombine is known as the carrier lifetime, τ. The process of photon generation of electron-hole pairs can occur when the semiconductor is illuminated with photons that have an energy, Ephoton = hν = hc/λ, greater than or equal to Eg, the bandgap of the semiconductor. h is Planck's constant and

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Chapter 1 is equal to 6.63x10-34Js. ν is the frequency of the photon. λ is the wavelength of the photon. Any photons with energy less than Eg will pass through the semiconductor. I.e., the semiconductor is transparent at wavelengths less than its bandgap. Silicon, which has a bandgap of 1.12eV, is transparent to photons with a wavelength of less than 1.1µm, which is in the infrared wavelength region. Thus, Si absorbs visible light and looks grey. Gallium phosphide, on the other hand, has bandgap of 2.3eV. Its bandgap is equal to the energy of green light and, thus, absorbs green and blue light. Therefore, gallium phosphide appears to be orange and somewhat transparent as red and yellow light passes through the semiconductor without being absorbed. Table 1 is a list of the bandgaps of several semiconductors. Included in this list is diamond. Generally, diamond is considered to be an insulator. As can be observed, its bandgap energy is considerably larger than that of the other semiconductors listed. However, in special situations, diamond can be a semiconductor. The general trend for bandgaps is that the larger (and heavier) the atoms are, the smaller the bandgap of the semiconductor. Thus, Si (28amu) has a larger bandgap than Ge (73amu). The bandgap is dependent on a number of factors including the strength of the bond between atoms in the crystal and the spacing between each atom (or length of each bond). As the temperature of the crystal increases, the bond length increases and the strength of the bond decreases. So, the bandgap is a function of temperature of the semiconductor, decreasing with increasing temperature.. A general formula for the bandgap as a function of temperature for any semiconductor follows the form of: Eg(T) = Eg(0K) – A T2 (B+T) Therefore, the lowest energy photon that the semiconductor can absorb changes with the temperature of the semiconductor. This change in bandgap energy as a function of temperature will also be important in understanding why the bias voltages of diodes and transistors vary with temperature.

Table 1 - Bandgap Energy Semiconductor Silicon Gallium Arsenide Germanium Indium Phosphide Diamond

Bandgap at 300K 1.12eV 1.424eV 0.661eV 1.35eV 5.46eV

Semiconductors

Bandgap at 77K 1.17eV 1.508eV 0.733eV 1.414eV 5.48eV

14

Chapter 1 Typical applications that rely on photogenerated carriers are in opto-isolators and optointerrupters where the light source is turned on and off or where an object interrupts the light beam. The speed of detection and extinction is limited by the lifetimes of the carriers, typically nanoseconds to a few microseconds.

CARRIERS IN INTRINSIC SEMICONDUCTORS In semiconductors, it is conventional to use the symbol n for the electron density, the number of electrons per cm3, and p for the hole density or the number of holes per cm3. In the intrinsic semiconductor, holes and electrons are created in pairs, n = p and the common symbol is usually designated ni, where the subscript indicates intrinsic material, or pure silicon. ni is also known as the intrinsic carrier concentration. ni2 = np (14) The number of thermally generated electron-hole pairs is a function of the semiconductor material and its temperature. ni = (NcNv)1/2 exp(-Eg/2kT)

(15)

where: (16)

(17)

me and mh are the effective mass of the electron and hole in the semiconductor, respectively. Strangely enough, these values can be very different from the mass of an electron in vacuum, mo = 9.1x 10-31 kg. The effective mass for electrons and holes for a number of semiconductors are given in Table 2.

Semiconductor Silicon Germanium Gallium Arsenide

Table 2 Electron Effective Mass, me 1.18 mo 0.55 mo 0.063 mo

Semiconductors

Hole Effective Mass, mh 0.81 mo 0.36 mo 0.48 mo

15

Chapter 1

Temperature Dependence of Hole-Electron Concentration As the temperature increases, the electrons get more energetic and the probability that an electron is excited into a higher energy state, the conduction band, is increased. Thus, the number of electrons in the conduction band is increased and the number of hole-electron pairs is increased. Finally, the conductivity is increased. The equation describing ni specifically for intrinsic silicon is 3 −5797 E g ( ) 15 2 T ni = pi = 4.8 x10 T e / cm3 T is the temperature in degrees K, and Eg = 1.12 eV, the energy gap in silicon. This equation gives ni = 1.5x1010 per cm3 at room temperature, 300 degrees K, or 27 degrees C. At 60 degrees C, ni = 1011 per cm3 We can use this characteristic to build temperature sensitive devices to measure temperature or to compensate for temperature changes such as thermistors. CURRENT IN INTRINSIC SEMICONDUCTORS Conduction due to the electrons is Jn = qnµnE

(18)

where the subscript n indicates electrons. Similarly, the hole conduction is Jp = qpµpE

(19)

where this time, the subscript p indicates holes. As the different physical properties of the electron and hole are different, the mobilities and effective masses of the electrons and holes are not the same. Therefore, the hole current and electron current need to be calculated separately and then added together. The total current density is J = Jn + Jp = qnµnE + qpµpE = q(nµn + pµp)E = qni(µn + µp)E

(20) (21)

Conductivity in intrinsic material is σi = qni(µn + µp) [Be careful here, this equation applies only for intrinsic material

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Chapter 1 where n = p = ni ]

(22)

It is of interest at this time to calculate the conductivity of intrinsic silicon and compare it to the conductivity of aluminum. Example: Calculation of conductivity of intrinsic silicon. Several properties of silicon at 300K (room temperature) are given in Table 3. From these properties we see that the intrinsic carrier density is 1.45x 1010 carriers per cm3, the mobilities of electrons and holes are 1500 and 475 cm2/Vs, respectively. Conductivity is then σi = 1.602 x 10-19 {coul}X 1.45 x 1010 {cm-3} X ( 1500 + 475) {cm2/Vs} = 4.59 x 10-6 (Ω cm)-1

TABLE 3 PROPERTIES OF SILICON Atomic Number Atomic Weight Density (gm/cm3) Atoms/cm3 ρi at 300o K (Ωcm) Dielectric constant εr

14 28.1 2.33 5.0x1022 2.30x105 11.7

µn at 300o K (cm2/Vs) µp at 300o K (cm2/Vs) ni at 300o K (cm-3) Dn at 300o K (cm2/s) Dp at 300o K (cm2/s) Electron affinity χe

1500 475 1.45x1010 34 13 4.05eV

Compare this with the conductivity of aluminum at 2.9 x 105 (Ω cm)-1. Obviously, intrinsic silicon is not nearly as good a conductor as aluminum. We will see shortly, however, that adding impurities to silicon can change its conductivity by several orders of magnitude. EXTRINSIC SEMICONDUCTORS In order to increase the number of free electrons or holes in a semiconductor, impurities are introduced. There are two types of impurities used, donors and acceptors. In Si, donor atoms typically have 5 valence electrons. In the crystal structure, four are bound in covalent bonds with adjacent silicon atoms. The remaining valence electron is easily disassociated, thus donating an electron for conduction. Only about 5 to 50meV are required to remove these remaining valence electrons. Since the amount of thermal

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17

Chapter 1 energy available is equal to kT, where k is Boltzman's constant (8.6x10-5eV/K) and T is temperature in Kelvin. At room temperature (~300K), the kinetic energy available to the system is ~25meV, which is greater than the average energy required to ionize an impurity in a semiconductor. Thus, essentially all of the extra electrons on the donor atoms are free conduction electrons at room temperature. Acceptor atoms usually have 3 valence electrons, and in the crystal structure, leave a valence vacancy which can trap an electron, thereby creating a hole for conduction. The energy required for this is typically slightly more than what is required to ionize a donor 10-100meV. However, most of the acceptors are ionized (created a hole) when the temperature of the semiconductor is above 100K. Pictorial examples are shown in Figure 8. A periodic table which identified the possible donor and acceptor atoms in IV semiconductors is below. Silicon with an excess of free electrons is called n-type while silicon with an excess of holes is called p-type. Typical donors are pentavalent atoms: antimony, phosphorous, and arsenic. Typical acceptors are trivalent atoms: boron, gallium, and indium. In GaAs and other III-V semiconductors, column II atoms are acceptors and column VI atoms are donors. Column VI atoms in GaAs are donors if they occupy a Ga site and are an acceptor if they occupy an As site. Thus, Column IV atoms (e.g., Si and Ge) in III-V semiconductors are called amphoteric dopants, meaning that they can act as either a donor or an acceptor.

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Chapter 1

Periodic Table

These impurities are introduced into the silicon either during crystal growth or after a crystal of pure silicon has been grown and sliced into wafers. Silicon wafers are grown by melting a large quantity of noncrystalline silicon in a carbon crucible. The melting point of silicon is 1460oC. To dope the silicon with impurities, an appropriate amount of donor or acceptor atoms are mixed into the melted silicon. A small single crystal of silicon, called a seed crystal, is then lowered slowly into the silicon melt, but pulled out before it completely melts. Because of silicon's high surface tension, some of the liquid silicon is pulled out with the seed crystal and begins to cool and solidify. The solidified silicon using the structure of the seed crystal as a template and, thus, grows as a single crystal. The seed crystal is continued to be pulled away from the silicon melt such that there is alway some liquid silicon wetting the solidified silicon. The solidified silicon is called a boule. Once the growth process is complete, the boule is sliced into wafers, which are then polished to remove the mechanical damage from the sawing process. The impurities can also be selectively introduced into the silicon wafer via diffusion or ion implantation. Typically, the wafer is allowed to oxidize to create a surface film of silicon dioxide, SiO2. Then some of the SiO2 is etched away in places where n-type or p-

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Chapter 1 type regions are wanted. During a diffusion, the wafers are heated and exposed to high concentrations of the desired impurity. The impurity diffuses through the openings in the SiO2 into the silicon displacing an occasional silicon atom in the crystal lattice. Upon cooling, the impurity atoms freeze in place, creating the n- or p-type regions. In ion implantation, a beam of high energy ions (ionized donor or acceptor atoms) bombard the surface of the silicon wafer. The SiO2 prevents the ions from reaching the silicon surface. However, they penetrate the silicon where ever the SiO2 has been removed. The silicon wafer is then heated to 700-1000oC to give the ions enough energy to incorporate into the silicon lattice. Only those impurities that occupy a site in the crystal lattice where a silicon atom would have been act as donors or acceptors and are called substitutional impurities. Those impurities that are in the silicon crystal but do not sit on a silicon site do not contribute a electron or hole to the semiconductor and are call interstitial impurities. Interstitial impurities, though not electrically active, reduce the mobility of the electrons and holes in the semiconductor as they will act as scattering sites as the electrons and holes move through the crystal Carrier Concentrations In intrinsic material, hole-electron pairs are thermally generated. These hole-electron pairs can also recombine. This process of generation-annihilation goes on continuously with the number of pairs at some equilibrium, where the number of electrons equals the number of holes. The average amount of time a hole or electron is free is called the carrier lifetime, τ. In n-type extrinsic material, there is no hole corresponding to the donated electron. There are still thermally generated hole-electron pairs, but the equilibrium is shifted so that there are fewer thermally generated carriers. Within the extrinsic semiconductors, two processes take place which contribute to the carrier concentration. Hole-electron pairs are created by thermal agitation just as in the intrinsic case. In addition, the impurity atoms are virtually all ionized creating the corresponding carriers. Usually, the doping level is high enough that the extrinsically provided carriers predominate over the thermally generated carriers. If both types of impurities are present simultaneously, they simply cancel each other out; an acceptor ion captures the excess electron provided by a donor ion. This is called compensation. Only the excess impurity concentration contributes to the conductivity. In most cases, one type of impurity predominates, creating an extrinsic material of the desired type. In perfectly compensated material, the concentration of electrons equals the concentration of holes and both are equal to the intrinsic carrier concentration, ni. But, the mobility of the electrons and holes is lower than in intrinsic material because of the larger number of impurity ion to scatter with.

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Chapter 1 Because of the physics of hole-electron pair formation, the total number of free carriers in the semiconductor is controlled by the mass-action law, equation 16 which is np = ni2. Thus, in an n-type material, the number of electrons is increased due to the donor ions, thereby causing the number of free holes to be reduced. In the n-type material, electrons are the majority carrier. Similarly in p-type material, holes are the majority carrier. Let ND represent the concentration of donor ions and let NA represent the acceptor concentration. The material must maintain charge neutrality so that the total number of positive charges will equal the total number of negative charges. ND + p = NA + n

(23)

Donors give up electrons so they become a positively charged ion. Ionized acceptors have a negative charge as they have captured an extra electron. In an uncompensated ntype material, NA = 0 and because virtually all the donors will be ionized at most temperatures, ND + p = n

(24)

Here the free electrons, n, have two sources; donors, ND, and thermally created holeelectron pairs, p. In all but extremely lightly doped materials, the number of impurity atoms is a much greater in concentration than is the intrinsic electron concentration, ni. In other words, ND >> ni. Thus, n ≅ ND. We can find the hole concentration from the mass action law (np = ni2) p = ni2/n ≅ ni2/ND

(25)

Likewise, when there is p-type semiconductor, the number of holes is determined by the number of acceptor atoms in the material. Thus, p ≅ NA. Again, we can find the electron concentration from the mass action law n = ni2/p ≅ ni2/NA

(26)

Now let’s look at the conductivity of a doped n-type material.

Example: Conductivity of doped material Assume silicon is doped with donors so that ND = 5 x 1012 atoms/cm3. Determine the conductivity of the material. The electron density n ≈ ND and we can find the hole density from the mass action law. p = (1.45 x 1010)2/5 x 1012 = 4.21 x 107 /cm3

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Chapter 1

(Note that p = 4 x 107 are all thermally generated holes. There are an equal number of thermally generated electrons, but this number of thermally generated electrons is insignificant when compared to the number of donors that contributed electrons.)

The conductivity then is σ = q(nµn + pµp) However, the added donors do more than increase the number of electrons in the material, they also decrease the mobility of the electrons and holes. This is because they act as Coulombic (or ionic) scattering sites (see the figure in the discussion on electron mobility in metals). The donor atons also slightly distort the unit cell because they are not exactly same size as the silicon atom they replace and the bonds that they form with the adjacent silicon atoms are not the same length as the Si-Si bonds.

The figure above shows the relationship between mobility and doping levels for both ntype and p-type silicon. For the doping level in the example, the donor ions do not significantly change the mobility of the carriers. Therefore, the the mobility of the holes and electrons are equal to the value of intrinsic silicon. The lower values of mobility would need to be used when calculating the conductivity of more heavily doped material, 1015 cm-3 to 1018 cm-3, which are typical doping levels for diodes and transistors. Thus, for our example, σ = 1.602 x 10-19 (5 x 1012 X 1500 + 4.21 x 107 X475) σ = 1.602 x 10-19 ( 7.5 x 10 15 + 2.00 x 1010) = 1.2 x10-3 (Ωcm)-1

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Chapter 1 This conductivity is almost 3 orders of magnitude greater than for intrinsic silicon. Note that the electron density is five orders of magnitude greater than the hole density which means that the holes contribute very little to the conductivity. It can be seen from this example that if the doping level is significantly higher than the intrinsic electron/hole density, the majority carrier does most of the conducting in the material and that the conductivity is directly proportional to the doping level. Fermi Level When drawing a energy band diagram, we indicate what type the material is (n type or p type) by the placement of the Fermi level, EF. The Fermi level is an imaginary energy level. When it is positioned halfway in between the conduction band and valence band, the material is intrinsically doped or evenly compensated. When the Fermi level is close to the conduction band, the material is n-type. The closer the Fermi level is to the conduction band, the more heavily doped is the material - the larger ND is. When the Fermi level is close to the valence band, the material is p-type. The closer the Fermi level is to the valence band, the more heavily doped is the material - the larger NA is.

No matter what the temperature of the semiconductor is, the Fermi level is halfway between the conduction and valence bands for an intrinsic semiconductor. However as the temperature increases, the Fermi level in a doped semiconductor moves towards the middle of the bandgap. The reason for this is that the intrinsic carrier concentration increases significantly with temperature. Thus, the extrinsic carrier concentration becomes less important as the temperature increases. Another way to express this is that as silicon becomes intrinsic as the temperature increases because ni > ND or NA. This is one reason why there is a maximum temperature of operation for most semiconductor devices. CONDUCTION BY DIFFUSION So far we have talked exclusively about conduction by drift of carriers in an electric field. However, conduction by diffusion is important in some cases. Diffusion result from the natural tendency of most particles move from regions of high concentration to regions of

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Chapter 1 lower concentration. That is, diffusion is driven by concentration gradients while drift is driven by electric fields. Everyone has experienced an example of diffusion where a person opens a bottle of perfume (or a package of fish), in a few minutes, a different person in another part of the room can smell it. The reason this diffusion happens is illustrated in Figure 9. In this figure, we have simplified the situation by assuming only four possible directions of motion: up, down, right, and left. In random motion, one fourth of the particles will be going in each direction. Now the left side of the volume has one half the density of the right side. Thus, there are twice as many particles going left across the boundary as there are going right in any period of time. In a short time then, there will be more particles on the left than as shown in the diagram. After while, the two sides will have the same density and the diffusion current will stop. The random motions will average out on both sides.

For n-type material, the diffusion current is given by J n = qD n

dn dx

(27)

where Dn is the diffusion constant. Note that if dn/dx has a positive slope (rises to the right), the concentration is higher to the right and electrons will diffuse to the lower concentration, to the left. Because the electrons have a negative charge, the current will be to the right, or in the positive x direction. Thus, if we look at diffusion current in a ptype material, J p = − qD p

dp dx

(28)

Here the negative sign reflects the fact that the holes will diffuse to the left causing current in the negative x direction. The diffusion constant of a hole or electron is related to that carrier's mobility by the

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Chapter 1 following equation: D/µ = kT/q

(29)

If we have both an electric field and a concentration gradient, the total current density is the sum of the drift current and the diffusion current densities. J n = qµ n nE + qD n

dn dx

for electrons

(30)

J p = qµ p pE − qD p

dp dx

for holes

(31)

and

The total current density is J = Jn + Jp dp dn − Dp ) dx dx (Diffusion)

= q (nµ n + pµ p ) E + q ( Dn (Drift)

(32)

THE pn JUNCTION We have now set the stage for looking at the p-n junction, the basis for most semiconductor devices. Figure 10 shows blocks of n-type and p-type material. In these drawings, only the donor and acceptor ions are represented along with the resulting free electrons and holes, the majority carriers in each material. The minority carriers, the holes in the n-type material and the electrons in the p-type materials, are not shown here to keep from cluttering up the diagram, especially since there are so few of them .

In each block in Figure 10, the positive and negative charges balance. Now if we bring those blocks together as shown in Figure 11, we still have charge neutrality, but we have

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25

Chapter 1 a concentration step dislocation. We have a high concentration of electrons to the left of the junction and a high concentration of holes to the right.

The donor and acceptor ions are fixed by the crystal lattice, but the holes and electrons are free to move. These carriers diffuse to the regions of lower concentration until the device comes to equilibrium shown in Figure 12.

In Figure 12, some of the electrons diffused over into the p-type material and some of the holes diffused over into the n-type material. These become minority carriers and are not shown. In fact, outside the depletion region, most of the excess minority carriers recombine with majority carriers and thus, are annihilated. Obviously, the carriers closest to the junction were the first to migrate by diffusion. This migration left behind some charged ions in the area marked as the depletion region. These charged ions create an electric field with the positive side in the n-type material and negative side in the p-type material. This field is essentially a potential hill which repels further migration. The carriers left behind, beyond the depletion region, still have thermal energy and move randomly in the area. If an electron had enough energy to overcome the potential barrier, and was moving in the proper direction, it would pass across the junction. This electron

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26

Chapter 1 would leave another charged ion behind, increasing the potential barrier, making it harder for others to cross. Now going back to the minority carriers. These are not shown in the diagram, but they too are distributed throughout their respective areas outside the depletion region. This distribution means that some are near the electric field at the edge of the depletion region, but instead of being repelled by it, they are attracted by it and are pulled across the junction. This act has the opposite effect on the barrier height, essentially neutralizing an ion, and lowering the barrier. The result is a continuous motion of charges back and forth across the junction at equilibrium. Minority carriers diffuse to the junction, are swept across the depletion region. This causes the potential barrier to be lowered which allows majority carriers to diffuse against the barrier. The charge motion is random, with the total current balancing to zero. However, at any one instant in time, there is a measurable current flowing across the p-n junction, even with zero voltage applied. The currents are quite small compared to other currents we will be looking at later. However, these currents are important in the operation of many devices. For example, these currents contribute to the dark current of a zero biased photodetector. It is often of interest to consider what happens if the doping levels are not the same on both sides of the junction. First, we note that charge neutrality in the device must be maintained and in fact is maintained both in the depletion region and in the non-depleted regions of the n-type and p-type materials. The depletion region is charge neutral, with the positive ions balancing the negative ions. Thus, if one of the regions has a higher concentration of doping ions, that part of the depletion region will be narrower than that in the region with lower doping. Similarly, if both regions had higher doping, the depletion would be narrower and the field gradient or field strength would be higher although the total potential difference would be the same. The depletion width, W, can be calculated using the following equation:

(33) where εr is the relative dielectric constant which is between 10-12 for most semiconductors, εo is the permittivity of free space (8.85 x10-14 F/cm), Vbi is the built-in voltage of the p-n junction, and Va is any externally applied voltage across the p-n junction. The built-in voltage of the p-n junction is equal to the electric field developed across the depletion width times the width of the depletion layer. It is equal to:

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Chapter 1

(34) One can see how the built-in voltage is developed when a p-type region is in contact with an n-type region by using the energy diagrams and the concept of continuity of the Fermi level. Continuity of the Fermi level means that, at zero bias, we redraw the energy diagrams of the p-type and n-type regions so that the Fermi levels of the two materials are at equal positions. Thus, the electrons in the n-type material need to gain enough energy, about Vbi, to overcome the energy barrier, the "hill" in the energy diagram, in order to move into the p-type material. And, the holes also need to gain energy to flow into the ntype material. [Increasing energy for a hole is negative energy.]

As can be seen from equation 33, as the voltage across the junction increases (i.e, the junction is reverse biased, Va < 0V), the depletion region width increases as well. Furthermore, the Fermi level of the p-type material is no longer drawn at the same energy as the Fermi level as the n-type material and the energy barrier for the holes to flow into the n-type material and electrons to flow into the p-type material increases. Changes in

the width of the depletion region will result in charge transfers, similar to those in capacitors. This phenomenon give s rise to the concept of junction capacitance, Cj. Cj = εr εo A/W

(35)

where A is the cross-sectional area of the p-n junction. At higher doping levels, the depletion width is thinner and we get a higher capacitance at

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Chapter 1 the junction. In some cases, this higher capacitance can cause problems with device speeds when there are large voltage changes, but also the narrow depletion region means carriers crossing the region take less time (the transit time, tt, is small) and that switching speeds can be increased if the change in voltage is low. In general, there is a speed/current trade-off in the design of these junctions. We will see the results of this trade-off later in this course. The design of semiconductor devices is a fascinating subject, but requires material beyond the scope of this course. In forward bias, Va > 0V, the depletion region decreases, the junction capacitance increases, and the electrons in the n-region need less excess energy to flow into the pregion/the holes in the p region needs less excess energy to flow into the n-region. From the discussion thus far, you would expect that when the externally applied voltage, Va, is equal to the built-in voltage, Vbi, the junction capacitance becomes infinite. But, of course, this doesn't happen because the resistance of the junction becomes zero, which shorts the junction capacitor. DIODES A pn diode is the simplest junction semiconductor device. A simple representation of a a p-n junction is shown below. At the junction, a depletion region forms where all free carriers have gone away, leaving the fixed donor and acceptor ions which cause a static field. This static field just balances the tendency of the majority carriers on either side to diffuse into the depletion region.

Diode symbol

Diode representation

In the steady-state case, there are several processes going on all the time that result in zero net current flow. (1) Majority carriers, holes from the p region and electrons from the n region diffuse into the depletion region. A few are energetic enough to cross the potential barrier, Vbi, and result in a current flow. Balancing this flow, (2) minority carriers, holes in the n region and electrons in the p region diffuse into the depletion region. The field sweeps them across the region into their respective majority regions. (3) Hole-electron pairs are continuously created in the depletion region by thermal agitation. These carriers are swept by the field from the depletion region. (4) Recombination of hole-electron pairs balances the thermal generation to maintain charge neutrality. The previous discussion has assumed no external connections. Let us now consider what

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Chapter 1 happens when a short circuiting wire is connected externally to the two ends of the device. Actually, the experiment is a big flop, nothing happens. A little thought will reveal to us that while there is contact potential difference between the two different semiconductor materials, there is also a contact potential difference between the wire and the semiconductor materials at both ends. It should come as no surprise that these contact potentials cancel each other; the sum of the contact potentials around the circuit is zero. If a forward voltage is applied to the diode, positive to the p material and negative to the n material, the potential barrier is decreased - by the value of the applied voltage if we assume that there are no resistive voltage drops in the p and n regions. Then, majority carriers diffusing into the depletion region can more easily overcome the potential barrier. Because both holes and electrons are involved and have opposite charges, the result is a net current flow from left to right. In many cases, the doping levels will be unbalanced (either ND>>NA or NA>>ND) such that either the hole current or electron current will dominate and the other will be ignored. If a reverse voltage is applied to the diode, the additional voltage across the depletion region will widen it. There will also be a very small reverse current, called the reverse saturation current. This current arises from the minority carriers diffusing to the depletion region and being swept across. This current is no longer balanced by forward diffusion of majority carriers because the potential barrier is now much higher and is maintained by the external source. (36) where Dn is the diffusion constant of minority electrons in the p region, Ln is the diffusion length of these minority electrons, Dp is the diffusion constant of minority holes in the n region, and Lp is the diffusion length of these minority holes. The minority carrier diffusion length is related to both the minority carrier diffusion constant and the lifetime of these minority carriers, τ. L = (D τ)1/2

(37)

It is the change in potential that is responsible for the difference in resistance of a diode when measured using a digital multimeter. Try this yourself, measure the resistance of a diode with the probes connected in one way. Then, measure the resistance again after switching the probes to the opposite terminals of the diode. One measurement will be very high, usually 1-10MΩ. The other resistance measurement will be much lower. The

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Chapter 1 reason for this is that the multimeter applies a very small voltage to the component under test. In one probe configuration, the diode is slightly reverse biased and a extremely small amount of current flows. In the other configuration, the diode is slightly forward biased and a much larger current flows. If we are going to use diodes in circuits, we must have some way to relate the voltage to the current. We will not discuss the physical electronics here, but it can be shown theoretically that the current is an exponential function of the voltage (38) where Is is the reverse saturation current, vd is the voltage across the diode in the forward direction shown earlier, and vT is the thermal voltage kT vT = (39) q where in this case, k is Boltzman's constant, T is the temperature in oK, and q is the electronic charge. We normally just use vT in voltage form for simplicity. At room temperature, 300 oK, vT = 25.9 mV. A plot of the diode equation is given in Figure 13. Note the change in scale for the forward and reverse currents.

The current flows easily when the applied voltage is equal to the built-in voltage, a point known as the flat band condition - where the conduction band energies and the valence band energies of the p region are equal to that of the n region. At this applied voltage, the depletion width is equal to zero. The capacitance is now dominated to another capacitance, known as the diffusion capacitance. This is a capacitance associated with the excess minority carriers that are injected into the p and n regions. In order to switch the diode off, these excess minority carriers must be removed from the layers in which they have been injected. Thus, there is a transient current, trr, that flow as the applied voltage on a diode is switched from Va > Vbi to Va < 0V and the minority electrons that

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31

Chapter 1 were injected into the p region and the minority holes that were injected into the n region return to the n region and p region, respectively. Non-Ideal Factors That Affect Terminal Characteristics Because of many factors that arise during manufacturing, the measured characteristic is somewhat different. Nevertheless, for moderate current densities, the diode equation serves quite well. There are significant differences at low currents. One of the most significant is caused by surface imperfections that cause Is to be much larger than predicted. Also, Is is a function of temperature which can cause significant reverse leakage currents at high temperatures due to the strong temperature dependence of ni and of the mobility of the minority carriers. At very high current densities, the resistance of the bulk semiconductor materials can have significant voltage drops that must be added to the junction characteristic to obtain the terminal characteristic. Lastly, as the reverse bias voltage is increased, at some point the p-n junction breaks down and a large amount of current flows. This reverse voltage is called the breakdown voltage, VBR, and is a function of the doping levels, NA and ND. The closer the doping levels are to ni, the larger VBR is. Similarly to the electron mobility in metals, the mobility of the electrons and holes in a semiconductor also have the same trend with temperature, initially increasing and then decreasing as temperature is increased. However, the relationship between resistance and temperature isn’t quite as straight-forward due to the fact that the intrinsic carrier concentration, ni, has an extremely strong dependence on temperature. To further complicate the calculation of the resistance as a function of temperature, the bandgap of the semiconductor, Eg, is, also, a strong function of temperature. The general formula for the change in bandgap as a function of temperature is Eg(T) = Eg( 0K) – A T2 (B+T)

(40)

For Si, Eg(0K) is 1.17eV, A is equal to 4.73 x 10-4 and B is equal to 636K. T, the temperature, is in degrees Kelvin. As can be seen from the formula, the bandgap of any semiconductor decreases with increasing temperature. This complicates the calculation of ni as the bandgap energy is a term in the equation (see eq. 15), with the result that ni increases even faster with increasing temperature. Because ni increases rapidly with temperature – faster than the mobility decreases with temperature, the resistance of a semiconductor decreases with increasing temperature. All of the thermal changes in various material properties of silicon impact the performace of a silicon p-n junction. For example, the change in the turn-on voltage (Vbi) for silicon

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Chapter 1 diodes is approximately –2mV/oC for changes near room temperature. Other device parameters are similarly affected by a change in temperature. The resistance of a silicon diode decreases with temperature because both the series resistance (the sum of the resistance of the bulk p-type and n-type semiconductor layers) and the junction resistance (which is inversely related to the rate of thermal generation of carriers) decrease with increasing temperature. The junction capacitance decreases with increasing temperature because the materials become less extrinsically doped as the intrinsic carrier concentration increases with temperature. When designing circuits that need to work in all types of environments – not just in the lab, engineers need to evaluate their designs to make sure that they work over the expected temperature ranges. All of these changes limit the maximum operating temperature of silicon devices. Generally, silicon devices are limited to well under 300oC operation. Larger bandgap semiconductors are less affected by temperature and, thus, can operate to higher temperatures. SiC devices have been operated up to 600oC. Diamond diodes are expected to be able to operate at far beyond this temperature. On the other hand, HgCdTe diodes, which have an extremely small bandgap, can not operate at room temperature, but must be operated at temperatures equal to that of liquid nitrogen (77K) or below. When you look at a data sheet, the operating temperature is specified. The manufacturer has guaranteed that the device or circuit will operate over this temperature range with the operating characteristics listed. Also, certain numerical assignments have been designated to identify the operating temperature range for some commonly used circuits. For example, digital logic circuits that begin with the numbers 74 are designed for commercial applications and generally can be operated from 0oC to 70oC. Whereas circuits that begin with the numbers 54 are designed to meet military specifications, the specified operation temperature range is –55oC to 125oC.

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Chapter 1 EXERCISES 1. An electron is released in the middle between two metal plates which are connected to the voltage source as shown. a. Draw an arrow beside the electron showing which way it will move. b. Draw an arrow showing the direction of current. c. If you stood on the grounded plate and threw the electron toward the other plate, how much energy would the electron initially have to have in order to reach the other plate?

2. An intrinsic silicon material is connected to a voltage source as shown. Draw arrows indicating the directions of motion for the holes and electrons. Which direction does current flow?

Holes

Electrons

Current

3. The doping level for an n-type silicon is 5x1015 donors/cm3. a. What is the density of electrons? n = _________ b. What is the density of holes? p = _________ c. What is the conductivity? σ = _________ 4. A pn junction has a reverse saturation current Io = 1 µA. A forward voltage of 0.65 volts is applied. What is the current? IF = _________ If a reverse voltage of 0.65 volts is applied, what is the current? IR = _______. 5. The symbol for a diode is shown along with a pn junction. a. Draw the polarity that will cause forward current in both devices. b. Draw an arrow showing the direction of forward current for both devices

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Chapter 1

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35

Chapter 1 Problems 1. Determine the conductivity of copper by using resistance and wire diameter in wire tables in the “Electrical Engineering Design Compendium”. 2. Determine the mobility of electrons in copper which has an atomic weight of 63.54, a density of 8.89x103 kg/m3, and one valence electron. 3. Two parallel plates 10 cm apart have a 10 volt DC voltage source connected between them. If you dropped an electron from the negative plate, how long would it take to reach the positive plate? If you stood on the positive plate and threw an electron toward the negative plate, what initial velocity is required for it to reach the negative plate? 4. What fraction of atoms in intrinsic silicon have broken co-valent bonds at 300oK? 5. You have a bar of intrinsic silicon that has a cross sectional area of 1 cm2 and is 10 cm long. a. What is the resistance between the ends? b. What is the resistance of an aluminum bar of the same dimensions? c. What is the current density in both bars if a 10 volt source is connected at the ends? d. What is the average drift velocity in each? 6. What doping level is required to make n-type silicon have a conductivity of 1 (Ωcm)-1? What would the doping level be if the material was p-type? 7. Suppose we could make wire of n-type silicon doped at 1015/cm2. What would the diameter of a 10 cm long piece of silicon wire have to be to have the same resistance as the same length of #24 copper wire? 8. For the resistor in Problem 7, what temperature range can this resistor be used if the design specification for the Si resistor is +/- 5% of the room temperature value? Ignore the change in mobility as a function of temperature. 9. Using Matlab, plot the bandgap of Si as a function of temperature from 0oC to 1200oC. a. At what temperature is the bandgap of Si equal to the bandgap of Ge at 300K? b. If you replaced the curve with a line between the temperatures of 0oC to 100oC, what is the slope of the line (the change in bandgap as a function of temperature)? 10. What is the junction capacitance (Cj) and built-in voltage (Vbi) of a Si diode that is doped

Semiconductors

36

Chapter 1 a. with NA = ND = 1x1015 cm-3? b. with NA = ND = 1x1015 cm-3? 11. Suppose that you have a piece of Si that is doped with arsenic to a concentration of ND = 1x1015 cm-3, how large a volume is needed to make sure that there is one Ar atom in it? Assume that the volume is a cube, what is the length of each side in nanometers? 12. A bar of semiconductor can be used as a propagation delay line. These are used on the inputs and outputs of various digital gates to minimize possible timing hazards that can arise when the signals arrive at a gate at slightly different times. For example, you might want to have the output from an OR gate ANDed with the output of an NAND gate. However, it takes 15ns for the output to appear on a 74LS32 OR gate and it takes 22ns for the output to appear on a 74LS00 NAND gate. a. What length of an n-type Si bar should be connected between the OR and AND gate so that the total propagation time (the time it takes for the output to appear on the OR gate plus the transit time of the carriers in the Si bar) equal the time that it takes for the output to appear on the NAND gate. Assume that the electric field across the semiconductor bar is 200V/cm and the mobility of the majority carrier is 1500 cm2/Vs. Please write the solution in µms. b. What is the voltage required to generate the 200V/cm across the n-type silicon?

Semiconductors

37

CHAPTER 2

DIODE CIRCUITS As discussed in the previous section, diodes are essentially one-way valves. They carry current in one direction, but block current in the other. In addition, in the forward conduction mode, the voltage must go above a threshold before significant conduction occurs. Diodes are used in a multitude of ways that utilize these characteristics. The objective of this course is not to look at hole and electron flows, but to see how the diodes can be used in circuits and how to analyze those circuits. For example, consider the circuit below, Figure 1. We are generally not interested in the holes and electrons in the diode, but rather, the current in the resistor or the voltage across the resistor.

Figure 1. A simple diode circuit NUMERICAL METHOD The diode being a non-linear element means that the basic methods of circuit analysis learned in your circuits course cannot be used. Then, how does one go about analyzing the circuit? For example, we can write a loop equation Vs = Vd + Vr = Vd + I*R Unfortunately, there are two unknowns in this equation, I and Vd. The relationship between voltage and current in the diode is given by I = I 0 (e

V ( D

VT

)

− 1)

Where VT is 25.9 mV at 300 degrees K, and IO is the reverse saturation current. If we put the two equations together we get VS = Vd + RI 0 (e

V ( D

VT

)

− 1)

(1)

This equation cannot be solved analytically. But it can be solved numerically. You can make a guess of the diode voltage and plug into the equation and see if you get a balance. Diode Circuits

1

If not, try another guess. This process can be carried out several ways. The simplest is to use a calculator, while more complex methods would use a computer. One interesting approach is to use a math solving program on a personal computer such as MathCAD. Literally all circuits could be solved this way, but that would be impractical, especially for more complex circuits. Secondly, the diode equation mentioned above only represents the hole-electron currents within the body of the diode. There is a considerable leakage on the surface of the semiconductor which makes the low current solutions in error. GRAPHICAL METHOD Instead of using the diode equation as the model, we can get quite satisfactory results by using simpler models, or by using graphical methods from the plotted V-I characteristic of the diode. Consider the circuit in Figure 2. The loop equation is Vs = Vd + I*R

(2)

a. b. Figure 2. Diode circuit to be solved graphically If we divide the circuit as shown in Figure 2b, the equation can be rewritten to solve for the voltage at the division. Vd = 5 - I*R

(3)

This equation can be solved for current I=

5 − Vd R

(4)

This equation represents the relationship between current and terminal voltage for the left-hand side of the circuit. The relationship between current and terminal voltage for the right-hand side is just the equation for diode. I = I 0 (e

Diode Circuits

V ( D

VT

)

− 1)

(5)

2

Each of these equations is just an I vs Vd equation and can be plotted on a graph. Equation 4 is a linear equation and its plot is called the load line. It can easily be plotted by selecting values for Vd and calculating I. For example, the two intercepts; Vd = 0 and I = 0 are convenient choices. The equation 5 is the diode I vs V characteristic for the diode. It is usually obtained from a curve tracer in the laboratory. Both of these equations are plotted on the same graph in Figure 3. The operating point for the circuit is where the two plots cross. V

( DV ) 5 − Vd T = I 0 (e − 1) R

(6)

Figure 3. Plots for graphical solution One problem with the graphical solution above is that the voltage scale is so large that precision in determining diode voltage is difficult. In many cases, the diode characteristic is plotted with a much lower maximum voltage than the voltage source in the circuit. In this case, the voltage intercept is off the scale and it is more difficult to plot the load line. This situation is easily handled by substituting in a fixed value of voltage in the load line equation and solving for the current. An example is shown in Figure 4. The maximum voltage on the diode characteristic plot is 2 Volts with the source voltage 5 Volts and a resistance of 1KΩ. Solving the load equation 5− 2 I= = 3mA R which gives us the current at Vd = 2 volts at the right hand end of the load line on the plot in Fig. 4. Figure 4. Diode Characteristic and Graphical Solution CIRCUIT MODELS The graphical method presented in the previous section is one possible method of solution of circuits with diodes. For obvious reasons, this method will get very tedious and time consuming for more complex circuits. What we would like to do is to find a way to solve the circuits analytically; with equations. We do this by using circuit models which are combinations of ideal circuit elements that behave the same or almost the same as the actual device. We then solve the circuit using standard methods of circuit analysis.

Diode Circuits

3

A model is what we use to represent an actual device. For example, we use a mathematical expression V=IR to represent the voltage across a resistor. This happens to be a very good model at low frequencies, below a few MHz. At higher frequencies, the inductive reactance becomes significant and the equation is no longer very accurate. Similarly, V = jωLI and I = jωCV are good models for inductors and capacitors at relatively low frequencies. We also use models for voltage and current sources. For example, a typical voltage source is far from ideal. It usually has internal resistance and cannot supply unlimited current. Now let us take a look at what these ideal circuit models represent. If we draw a plot of the relationship between current and voltage of a resistor, we get the plot in Figure 5a. Similarly if we plot the I-V characteristic of the ideal voltage source, we get the plot in Figure 5b. The I-V characteristic of the voltage source in series with the resistor is shown in Figure 5c.

a. Resistor b. Voltage source c. Series combination Figure 5. Models for a Resistor and a Voltage Source Circuit Models for Diodes. Now let's look at possible circuit models we can use for a diode. Clearly, a single straight line would not make a very good model. However, a crude approximation can be developed using two straight lines as shown in Figure 6a, a model of an ideal diode. The two straight lines are used only for a specific region as shown. The ideal diode is considered an open circuit if the voltage across it is less than zero, and is a short circuit if the forward current is greater than zero. These two regions are called the cutoff and Figure 6. Two Possible I-V Characteristic Models of a Diode conducting regions or the reverse biased and forward biased regions respectively. We will carry this nomenclature forward with us through more advanced models. The semiideal model in Figure 6b is a better model of the actual diode curve and is used

Diode Circuits

4

extensively. In this model I = 0 for V < Vγ, and V = Vγ for I>0, where Vγ is the threshold voltage. Another way to look at these models is to translate them into circuit elements. In fact, this is the way we will use these models for conceptualizing circuit operation. The circuit elements representing these models are shown in Figure 7. Figure 7. Circuit elements representing the models in Figure 6 SOLVING DIODE CIRCUITS USING DIODE CIRCUIT MODELS As a first example, let us try the solution of a simple rectifier circuit shown in Figure 8. We will do this first with an ideal diode model and then with the semi-ideal model. Solution for this circuit may mean finding the current or the voltage across the resistor or diode. Figure 8. Diode Rectifier Circuit Solution Using The Ideal Diode Model When attempting to solve this circuit assuming an ideal diode, the first thing you notice is that the diode has two regions of operation. How do you know which to use? The answer to that question comes more from experience than anything else. However, some quick mental analysis is often useful at arriving at a good starting point. If all else fails, simply guess and then check to see if the resulting solution is consistent with your original guess or assumption. For this example, a reasonable guess is that the diode is forward biased and is therefore conducting. In this region, for the ideal diode, V=0 and I>0. Putting this model into the circuit, as shown below, the resulting current is I = V/R = 2.5 mA Since I>0, our solution is consistent with the original assumption.

Figure 9. Rectifier Circuit With Diode Replaced by Ideal Model (Conducting) Solution Using The Semi-Ideal Model Now let us use the semi-ideal diode model and solve the rectifier circuit again. The model equations are: V=0.70 for I>0 and I=0 for V0. The circuit is redrawn below in Figure 10 with the diode replaced by the assumed model. Solving this circuit, I= (5.0 - 0.70)/2K = 2.15 mA. Again, since our solution is consistent with the circuit model, we can assume the solution is correct. Figure 10. Rectifier Circuit With Diode Replaced by Semi-Ideal Model (Conducting) We note that these two solutions are significantly different because we used significantly different models for the diode. The difference would have been smaller if the source voltage had been very large compared to Vγ. Example 2, A More Complex Circuit As a second example, let us look at the circuit below. We will use the semi-ideal diode model for this problem. At first glance it is not clear whether the diode is conducting or not. Thus, we are not given any guidance in selecting the appropriate model. Let us guess that the diode is conducting. In that case, the circuit is redrawn with the diode replaced by its model in the conducting region.

a. Original Circuit

b. Circuit With Conducting Model

Figure 11. Diode Circuit for Example 2 We can solve this circuit by simply determining the currents in the branches. I1 = (5.0-.70)/16K = .269mA I2 = .70/2.2k .318mA I 3 (The diode current) = .269-.318 = -.049 mA The result implies that the current in the diode is in the reverse direction which is inconsistent with the diode model chosen. Our original guess was wrong. Let's try this problem again with the model for the non-conducting or cutoff region. The circuit is shown below. Figure 12. Circuit of Example 2 with Non-Conducting Model Diode Circuits

6

Solving this circuit, we find that the voltage across the diode is: Vd = 5.0x 2.2k/(2.2K + 16K) = 0.604 V ( 1.70, The diode will be conducting. The circuit is shown below. For this case, Vout = 1.70

Diode Circuits

for Vin > 1.70.

7

Figure 15. Clipping Circuit With Diode Conducting If we combine the two regions of operation, and draw the waveforms generated by this circuit, we will get the results shown in Figure 16. Actually, the diode begins to conduct when the forward voltage is in the neighborhood of 0.60 volts and fully turns on at about 0.70 volts. In between is some sort of smooth transition. The sharp transitions of the waveforms in Figure 16 would actually be rounded. For the purposes of this course, this transition is not very important. Figure 16. Output Waveform of the Diode Clipping Circuit With A Sinusoidal Input Signal Advanced Models The models in Figure 6 are obviously only crude approximations to the actual curve. A more advanced model will take into consideration that the diode current is not zero below the knee of the curve. Looking just at that portion of the curve, it looks almost resistive, with a rather high resistance. In fact, this region is often modeled that way with a resistor on the order of 1 megohm. The fact of the matter is that a significant portion of the current in this region is due to surface leakage, (resistive). For the conducting portion of the curve, the curve is nearly vertical, but not quite. Again, this suggests another resistance, but in series with a voltage source. This resistance is mainly due to the body resistance of the bulk material. The result is a two-region linear model that is piece-wise continuous. The models and mathematical expressions are shown in Figure 17.

Figure 17. Two-Resistor Model of a Diode

Diode Circuits

8

The circuit model shown in Figure 17 is a much better representation than the earlier ones, but even this one differs greatly from the actual characteristic in the region of the knee. More sophisticated models would use three or more regions for the model. Choosing an Appropriate Model The model for the ideal diode is useful for many purposes that do not require great precision in circuit solution. The two-resistor model is rather complex and is sometimes used where relatively accurate analysis is required. However, this model is more difficult to use than the other two. As a compromise, we will use the semi-ideal model for most of the analysis in this course. It will give a reasonably good solution to digital switching circuits used while not adding unnecessarily to the circuit complexity. For silicon diodes, the value of Vγ is usually taken between 0.60 and 0.70 volts. There are physical justifications for the threshold voltage assumed, but the value used is often selected from observations of the typical voltage measured across the diode with normal operating current. For integrated circuit diodes, the cross sectional area of the diode body is kept rather small and the current density is often rather high. Thus, for integrated circuits, 0.70 volts is often used. For discrete diodes, the current densities are often somewhat lower, allowing a lower value, usually 0.65 volts. In both integrated circuits and low-power discrete circuits, any reverse voltage applied during operation is quite low, allowing the reverse current to be neglected. In power diodes, the current densities are often very high, approaching the upper device limit, and the reverse voltages also often approach the device limits. In these cases, the body resistance during forward conduction and the reverse leakage current during reverse biasing cannot be neglected. Thus, in this case, the two-resistor model is more appropriate and is often used. Germanium diodes are usually used only for low power applications, and usually used as discrete devices as germanium is rarely used in IC's. In practice, Vγ is usually taken between 0.2 and 0.3 volts. We will use a slight modification to the semi-ideal model for most of this course where we are talking about switching circuits. The model we will actually use is:

We will not operate where 0.600 I=0 for V0 I = 0 for VD < 0.700

12. A device found in the Martian ruins has a model described by the following equations. Equations: i.

Vx = 2.5 V for I > 25 mA

ii.

Vx = 100 I for -5.0 < Vx < 2.5 V

Diode Circuits

16

iii. I = -5.0 mA for Vx < -5 V a. Draw a circuit model for each region of operation. b. Draw the I-V characteristic for the device

Diode Circuits

17

CHAPTER 8

MOSFET Field effect transistors can be thought of as voltage controlled resistors for many applications. The physical construction of a typical enhancement MOSFET is shown in Figure 1. This figure represents an n-channel, enhancement mode device.

Figure 1. Construction of a MOSFET. (n-channel, enhancement), This device is built on a substrate of p-type silicon. Two heavily doped n-type tubs are diffused in for the source (S) and the drain (D). The channel region is covered with a silicon-dioxide, SiO2, insulating layer. A metal layer is deposited on top of the insulating layer to act as a gate (G). If we simply apply a voltage between the source and drain, one of the p-n junctions at either the source or drain would be reverse biased and the device would not conduct. The gate becomes the control element as discussed next. A typical common-source circuit connection is shown in Figure 2. In this case, both the drain and the gate are made more positive than the source, which is grounded.

Figure 2. Typical biasing connection. Ignoring the drain supply voltage, VDD, for a moment, let us consider the gate supply only as shown in Figure 3. If the gate voltage is made positive enough, a number of electrons from within the p-type substrate will be attracted to the gate area. Because the gate is insulated from the substrate, these electrons will simply accumulate just below the gate, effectively changing that area to an n-type material; hence an n-channel.

Figure 3. The MOSFET with a positive gate voltage.,

MOSFETs

1

As shown in Figure 3, this locally enhanced n-type region between the source and drain allows conduction between the source and drain terminals. Note that if the gate voltage is made even more positive, more electrons accumulate, the electron density increases and the cross-sectional area increases, lowering the channel resistance. Also, there is a threshold voltage, VT, below which not enough electrons will be attracted to turn the channel region into n-type material, and the channel will not conduct. This threshold voltage is a function of doping levels and thickness of the oxide layer. With an insufficient voltage on the gate to establish the channel region as n-type, there can be no conduction between the source and drain. But when the gate voltage is above the threshold, the channel is established. Then, if we apply a voltage between the drain and the source,VDS, we will get a current to flow. As long as we keep the drain-source voltage, low, the MOSFET acts just like a voltage controlled resistor. The resistance of the channel is controlled by the gate voltage. This region of operation is known as the triode region., Next, let's examine the voltage between the gate and the channel at three points along the channel as shown in Figure 4.

Figure 4. MOSFET under operation in the triode region.(VDS small), The voltage difference between the gate and the channel at point 1 is essentially the gatesource voltage, VG1 = VGS. Likewise, the voltage between the gate and the channel at point 3 is the voltage between the gate and drain, and VG3 = VGS - VDS = VGD. Note that VGS > VGD. Because the voltage between the gate and the channel at the drain end is less than the voltage between the gate and the channel at the source end, the channel is thinner and there is lower electron density in the channel and lower conductivity at the drain end. The voltage between the gate and middle of the channel is between the other two, VG1 >VG2 >VG3. Because the voltage between the gate and various points along the channel is not constant, the channel will not be a constant width. Figure 5 shows the channel in this condition. The channel-thinning effect becomes more pronounced as VDS is increased. In fact, if we increase the drain voltage high enough, the voltage between the gate and the channel near the drain will fall below the threshold voltage VT. One might think the current would stop, but instead the current simply remains constant as VDS increases further. This effect is called PINCH-OFF. If the current were actually cutoff, the channel would be isolated from the drain, and the gate-channel voltage would increase back up to

MOSFETs

2

the gate-source voltage, and conduction would start again. Pinchoff is simply an equilibrium point with constant current for a fixed gate-source voltage. Figure 6 shows the channel in pinchoff. Pinchoff begins when VGD < VT.

Figure 5. Channel in Triode

Figure 6. Channel in Pinchoff

The drain characteristic for a ZVN-3310, an enhancement-mode MOSFET is shown in Figure 7. The drain characteristic is a plot of drain current versus drain-source voltage at various gate-source voltages. If we held VGS constant, we would get a single curve, but an infinite number are possible, so we select an appropriate set to represent the device. For any particular curve, it starts at ID = 0 and VDS = 0. As VDS increases, the current increases almost linearly in the triode region. As VDS increases further, channel thinning occurs and the curve does not rise as rapidly. The rate of rise decreases until there is constant current in pinchoff.

Figure 7. Drain Characteristic for a ZVN-3310, n-channel MOSFET`

MOSFETs

3

MODELING EQUATIONS FOR THE MOSFET There are three distinct regions of operation: cutoff where the gate voltage is below threshold and no current flows, triode where the current is nearly a linear function of drain voltage, and pinchoff where the current is constant for fixed gate voltage. Figure 8 shows the voltage polarity and current direction definitions. Positive drain current enters the drain and leaves by the source. Gate current is zero because it is insulated from the channel. The order of the subscripts In VGS and VDS indicate a voltage drop from gate-source Figure 8. N-channel, enhancement MOSFET and drain-source respectively. CUTOFF REGION (VGS < VT) In this case, The gate voltage is not high enough to create a conducting channel so there is no conduction. This case can be represented by a line on the x-axis of the characteristic curve with ID = 0. TRIODE REGION (VGS -VT > VDS ) [Could also be written VGD >VT ], As long as VGS -VT >VDS, the voltage between the gate and every point in the channel is greater than the threshold voltage. Under these conditions, the approximate drain current is ID = k(2(VGS-VT)VDS-VDS2) Equation 1 Where k is given by: µε W k= 2 tL and: µ= carrier mobility, ε= dielectric constant, W = channel width, t = dielectric thickness, L= channel length., If Equation 1 is compared to the characteristic curves, you will note that Eq. 1 is the equation for a parabola, as seen in the curves where the curves leave the origin. At VGS -VT =VDS , Eq. 1 reaches its maximum value of Idmax = kVDS2 and has a slope of zero. You will also note that, at that voltage, the voltage between the gate and the channel at the drain end is just equal to VT; in other words, the threshold of pinchoff PINCHOFF (VGS -VT < VDS ) In this region, the voltage between the gate and some parts of the channel is less than the threshold voltage. The current will remain constant as VDS increases. ID = k(VGS - VT)2

MOSFETs

Equation 2

4

Note that this current is just the same as the point where VGS -VT =VDS from Eq. 1. Similar to the circuit models for all electronic devices, the model equations are approximations to the actual curves. For example, Equations 1 and 2 are plotted in Figure 9 for the case of VGS = 4 V, VT =2 V, and k=100 µA/V2 . Equation 1 is a parabola rising from the origin and curving back down. Equation 2 is simply a straight line with fixed current and not a function of the drain voltage. The two curves intersect at the peak of the parabola, the transition between triode and pinchoff operation. The parabola is valid only up to the intersection, and the straight line is valid only after the intersection, resulting in a composite similar to the actual characteristic curves in Figure 8.

MOSFET Characteristic, Vgs=3 VT=2, k= 100 microamp/v**2

Drain Current

400 Eq. 2

300 Eq. 1

200 100 0

0

1 0.5

2 1.5

3

4

5

2.5 3.5 4.5 5.5 Drain-Source Voltage

6

7 6.5

Figure 9. MOSFET Curve Using Modeling Equations., Figure 10 is a characteristic curve of an n-channel MOSFET using the models developed above. Also plotted on the graph is the boundary between the triode and pinchoff regions where VGS -VT =VDS. On this graph, VT = 2 V. This calculated characteristic curve is very much like actual characteristic curves taken on a curve tracer.

MOSFETs

5

Enhancement MOSFET Characteristic

Drain Current, microamps

2000

VT = 2.0 volts, k=100 microamps

Vgs=6 1500

1000

5

500

4 3

0

0

1 0.5

2 1.5

3 4 5 6 7 2.5 3.5 4.5 5.5 6.5 Drain-Source Voltage

Figure 10. Drain characteristic from model equations for an n-channel, enhancement mode MOSFET with VT = 2.0 volts and k = 100 µA/V2. CIRCUIT MODELS FOR THE MOSFET We have described three distinct regions of operation: cutoff, pinchoff, and triode. Each region can be represented by the equations and used to solve circuits as shown in following sections, or each region can be represented by circuit elements and circuits solved using standard circuit analysis techniques. Cutoff (VGS < VT), ID = 0 In this case we can represent the MOSFET as shown in Figure 11. There is no current between the drain and source as the channel is open. Thus, we represent the channel as an open circuit. We also represent the gate terminal as an open circuit recognizing that the gate is insulated from both other terminals by an insulating oxide layer. Pinchoff (VGS > VT and VGS-VT < VDS) ID = k(VGS - VT)2

Figure 11. MOSFET Cutoff circuit model.

In this case, current is only a function of gate voltage and thus, the current in the channel can be represented by a dependent current source as shown in Figure 12.

Figure 12. MOSFET Pinchoff model MOSFETs

Figure 13. MOSFET Triode model 6

Triode (VGS > VT and VGS-VT > VDS) ID = k(2(VGS - VT)VDS - VDS2) If we assume the VDS2 term can be ignored, or 2(VGS - VT) >> VDS, then ID ≈ 2k(VGS - VT)VDS and the term 2k(VGS - VT) has units of Siemens or Mhos, then the inverse is a resistance defined as RDSon. In other words, we can represent the drain-source channel as a resistor as shown in Figure 13. RDSon =

1 2 k (V GS − VT )

SOLVING MOSFET CIRCUITS To illustrate the use of models in solving MOSFET circuits, consider the circuit shown in Figure 14. In this circuit, VDD = 6 volts and we will solve the circuit for input voltages of Vin = 0,3, and 6 volts. Solution of these circuits almost always involves finding the output voltage, which in this case is VO = VDS, the voltage between the drain and ground, or the current in the drain, ID. This Figure 14. MOSFET inverter circuit. VT = 2 MOSFET is the same one whose volts and k = 100 microA/V2. calculated characteristic is shown in Figure 10. First, we will solve the circuit using the circuit models and then using the model equations. SOLUTION USING CIRCUIT MODELS Vin = 0 volts

MOSFETs

7

In this case, Vin = VGS = 0 volts. Because VGS is below the threshold voltage, VT, we will replace the MOSFET with its cutoff model as shown in Figure 15. From this circuit we see that ID = 0 and because there will be no voltage drop across the 5K resistor, VO = VDD = 6 volts. Figure 15. MOSFET inverter with cutoff model inserted.

This solution is consistent with the cutoff assumption: ID = 0 and VGS < VT.

Vin = 3 volts. The input voltage is clearly above the threshold, so the MOSFET must be conducting. Let us assume the device is in the pinchoff region. Then, we can replace the MOSFET by its pinchoff model as shown in Figure 16. The current in the dependent current source is ID = k(VGS - VT)2 = 0.1mA/V2(32)2 = 0.1 mA

Figure 16. MOSFET inverter in pinchoff.

The output voltage is VO = 6 - 0.1mA * 5 kOhms = 6 - 0.5 = 5.5 volts Now we must check to make sure we really are in pinchoff. (VGS - VT) = 1 volts which is less than VDS = VO = 5.5 volts. Therefore, we have met the conditions of pinchoff and our solution is valid. Vin = 6 volts Because we are now driving the gate to a higher voltage, let us assume the MOSFET is in triode. Then we can use the RDSon model for the device in the circuit as shown in Figure 17. RDSon = 2 k (V 1 −V ) = 1.25KOhm GS T

Solving the circuit then, MOSFETs

Figure 17. MOSFET inverter in triode 8

ID =

6 = 0.96mA 5K + 1.25K

and VO = 6 - ID* 5K = 6-4.8 = 1.20 volts Now checking to see that we are truly in the triode region, (VGS - VT) = 4 volts which is greater than VDS = VO = 1.20 volts. What if we had guessed the wrong region of operation? The cutoff region is easy because if Vin < VT, then (VGS - VT) is negative. But for instance we had assumed triode with Vin = 3 volts, then the circuit would be the same as in Figure 17 but with RDSon = 5 K Ohms, and VO = 3.0 volts which is greater than (VGS - VT) = 1 volt. In other words, our result would have been inconsistent with the assumption. In the third case with Vin = 6 volts, If we had assume pinchoff, ID = 1.6 mA and Vo = -2.0 volts, clearly an illegal result. (The dependent current source model is only valid for cases with VDS > (VGS - VT.) SOLUTION OF THE MOSFET INVERTER USING MODEL EQUATIONS We can also solve the circuit using the model equations for the three regions. Using the model equations, we don’t redraw the circuit for each case, but instead, we write an expression for Kirchoff’s voltage law around the drain circuit in Figure 18. 6 = 5K * ID + VDS Figure 18. MOSFET inverter circuit This equation has two unknowns, VDS and ID. We use the model equations to represent the relationship between the two. We will solve the circuit again for the same three cases using the model equations. Vin = 0 This case solves the same way as with circuit models. Since we are in cutoff, ID = 0, thus, VDS = VO = 6 volts. Vin = 3 volts

MOSFETs

9

Now we have to decide which region or model to use. If we assume pinchoff as in the previous method, then ID = 0.1mA/V2(VGS - VT)2 = 0.1(3-2)2 = 0.1 mA, and VO = 6 - 0.1mA*5K = 5.5 volts. Vin = 6 volts Assuming triode, ID = k(2(VGS - VT)VDS - VDS2). Since this equation is a function of VDS, we will have to use it to replace ID in Kirchoff’s equation for the circuit. 6 = R*k(2(VGS-VT)VDS-VDS2) +VDS The only unknown is VDS, but this equation is a quadratic. For our specific case, 0.5VDS2 -5VDS + 6 = 0 or VDS = 8.605, or 1.394 volts We get two possible solutions because our model equation is the equation for a parabola. However, only one solution fits our criteria for triode region, (VGS - VT) > VDS or VDS = 1.394 volts. But wait! This solution is not the same as the one we got using circuit models. In that case, we got VO = 1.2 volts. This discrepancy is a result of approximation we used in developing the RDSon circuit model where we assumed VDS VSD 2

ID = - k(VSG - VT)

Figure 22. p-channel MOSFET

Triode, VSG - VT < VSD ID = -k(2(VSG - VT)VSD - VSD2) We can also represent the p-channel MOSFET with the same circuit models as shown in Figure 23. Note the current reversal in the pinchoff model.

a. Cutoff b. Pinchoff Figure 23. Circuit models for the p-channel MOSFET

MOSFETs

c. Triode

13

INVERTER CIRCUIT WITH A P-CHANNEL MOSFET For an example, a p-channel MOSFET is used in an inverter circuit shown in Figure 24. In this circuit, the source is connected to the positive supply. Note that the drain current is negative so that actaul current goes from positive to negative. To avoid negative currents, we will use I = -ID. The source-gate voltage is not the input voltage, but the voltage difference between the VDD and the input VSG = VDD - Vin and the output voltage is the voltage across the resistor or VO = VDD - VSD. For the purposes of this example,VT = 1 volt, and k = 5 mA/V2. We will analyze this circuit for input voltages of Vin = 0, 3, and 5 volts. We will start with Vin = 5 volts.

Figure 24. P-channel MOSFET inverter circuit.

Vin = 5 volts In this case, VSG = 0 volts, below the threshold voltage, so the MOSFET is cutoff, ID = 0, and VO = 0 volts. Vin = 3 volts. Here, VSG = 2 volts, and if we assume pinchoff, I = -ID = 5(2-1)2 = 5 mA. VO = 2.5 volts. (2.5>(VSG - VT), so pinchoff is correct.) Vin = 0 volts. VSG = 5 volts. If we assume triode operation, we can replace the MOSFET with RDSon = 25 Ohms. Then VO = 5(500/525) = 4.76 volts. Checking our assumptions, VSD = 0.24 volts < VSG - VT), so triode is correct. Also, 2(VSG - VT) = 8 0.) Because VGS2 =VDS2 =VDD -Vo, then the equation of operation for Q2 is: ID2 = k2(VDD - Vo -VT2)2

Equation 6

For Q1, as in the previous example, VGS1=Vin, and VDS1 =Vo. Also, with no load, the current in the two devices is the same, ID1 =ID2 . Now Q1 may operate in cutoff, in the triode region, or in the pinchoff region.

Figure 25. MOSFET inverter with an active pull-up. Q1 CUTOFF, Cutoff, Vin VT1, and Vin -VT1 VT1 , and Vin -VT1 >Vo In this case, ID1 = k1[2(Vin - VT1)Vo - Vo2] = k2(VDD - Vo - VT2)2 which again is a quadratic equation which can be solved for Vo. Vo2(k1 + k2) - Vo[2k2(VDD - VT2) + 2k1(Vin - VT1)] + k2(VDD - VT2)2 = 0 Equation 8 Example: Both Q1 and Q2 are made on the same piece of silicon at the same time, making the threshold voltage, VT =VT1 =VT2 as well as making ε, µ, and t the same for both devices. However, Q2 is made much longer and narrower than Q1, making k2 much smaller than k1. For the circuit shown in Figure 25, let us assume VDD =10V, k1 =1mA/V2 , and k2 =0.1mA/V2 , and VT =2V. Plot Vo vs Vin. Solution: The solution must be taken in three parts: cutoff, pinchoff, and triode regions for Q1. Cutoff: Vin 2, and Vin -2Vo ) For all values of Vin >3.922, Q1 will be in the triode region, and Equation 8 must be solved. Vo2 (1.1) - Vo (2Vin -2.4) + 6.4 = 0, Because a solution to the quadratic equations will result in two answers, you must choose the one which conforms to the triode operation, Vin -2>Vo . A table of solutions is given below and the plot is shown in Figure 26. TABLE 1 Vo vs Vin for the active pull-up inverter example. Input voltage Output voltage 8.0

VT , and VGS -VT > VDS ⇒ Vin> VT, and Vin -VT >Vo ID = k(2(Vin -VT)Vo -Vo2 )

(6)

PINCH-OFF REGION, VGS > VT, and VGS -VT < VDS ⇒ Vin > VT, and Vin -VT VT, and VSG -VT > VSD ⇒ Vin < VDD - VT and Vin +VT VT, and VSG -VT Vo

CMOS Logic

2

ID = -k(VDD -Vin -VT)2

(9)

Now let us turn to calculating the voltage transfer characteristic. The characteristic is given in Figure 3 for arbitrary VDD>2VT. There are five cases or regions of the input voltage of interest. We will start with the input voltage low. CASE 1.

Vin < VT,

If we start with the input at 0 volts, The n-channel transistor will be off and the p-channel will be on. n-channel: Vin < VT The n-channel is cutoff and therefore, ID =0 p-channel:, Vin < VDD - VT, The p-channel is turned on and with no current because the n-channel is turned off. Equation 9 has no solution for ID =0 except VDD=Vin+ VT which is contrary to the input conditions. Therefore, the pchannel is not in pinchoff. Equation 8 does has a solution with ID = 0 at Vo=VDD with the p-channel in the triode region. This case holds for the input voltage up to Vin=VT as long as VDD>2VT. VDD 2 We start into this case at the boundary Vin just begins to exceed VT which makes the output voltage very close to VDD, leaving the p-channel in the triode region and the nchannel just beginning to conduct in the pinchoff region. These conditions hold throughout this case.

CASE 2. VT < Vin

VT , and Vin -VT < Vo ID = k(Vin -VT)2 p-channel: TRIODE REGION, Vin < VDD - VT and Vin +VT VT, and Vin -VT < Vo ID = k(Vin -VT)2 (13) p-channel: pinch-off, VDD -Vin > VT, and VDD -Vin - VT < VDD -Vo (Vin + VT > Vo) ID = -k(VDD -Vin -VT)2 (14) Setting the two currents equal,

Case 3. Vin =

CMOS Logic

4

k(Vin -VT)2 = k(VDD -Vin -VT)2

(15),

or, +/-(Vin -VT) = +/-(VDD -Vin -VT) This yields only a single value for Vin, V Vin = DD 2 In this case, the output voltage can take on any value in the range VDD V − VT ≤ Vo ≤ DD + VT 2 2 Case 4:

(16), (17),

(18),

VDD < Vin < VDD -VT 2

This case is the reverse of case 2. The output voltage has fallen low enough that the nchannel is in the triode region while the p-channel continues in pinchoff. n-channel: TRIODE REGION, Vin > VT, and Vin -VT > Vo ID = k(2(Vin -VT)Vo -Vo2)

(19)

p-channel: PINCH-OFF REGION, VDD -Vin > VT, and Vin + VT > Vo ID = -k(VDD -Vin -VT)2

(20)

k(2(Vin -VT)Vo -Vo2) = k(VDD -Vin -VT)2

(21)

As in case 2, the two currents can be set equal to each other and the resulting quadratic solved for any input voltage within the range of this case. The end-points of the curve in Figure 3 can easily be found by plugging in the two extreme values of Vin, Vin = 0.5VDD, or Vin = VDD - VT. The solutions are VO = 0.5VDD - VT, or VO = 0, respectively. CASE V. Vin > VDD -VT, In this case, the p-channel is off and the n-channel is in the triode region and Vo =0 V., CURRENT DURING SWITCHING When VT < Vin < VDD -VT, both transistors are conducting. Outside of this region, one of the transistors is cutoff and the current is zero. This current occurs during the transistion and produces power dissipation in the transistors. Obviously, the power dissipation will be higher as the number of transitions during a period of time increases. At very low speeds, this power loss is insignificant, but can become quite high at several megahertz. What does this current look like?, Because the currents in the two transistors are equal, it is easier to calculate the current for the transistor in the pinch-off region. Also, the circuit is symmetric so we only need CMOS Logic

5

to calculate the current for one half of the transition. During the first half of the transition, the n-channel is in the pinch-off region. The current is given by ID = k(Vin -VT)2

(22)

The current during the transition is given in Figure 4. Notice that the current graph varies as a square function up to VDD/2 and is symmetric about VDD/2 for identical MOSFETs.

Figure 4. Current through a CMOS inverter as a function of Vin. CMOS GATE TERMINAL SPECIFICATIONS, From our discussion so far, we have only described the operation of the inverter and developed the voltage transfer characteristic. We have not yet developed terminal specifications for the circuit. It is clear that an input voltage at either rail voltage will cause one of the transistors to turn off and with no load, the output voltage will be at the other rail. That situation prevails even if the input voltage comes away from the rail by as much as VT. To go beyond this point or to operate with a load, involves a complex relationship between voltage and current. The CMOS manufacturers have taken a different route. If we take a look at the voltage transfer characteristic, we see that the slope of the curve changes drastically during the transition. The slope is the ratio of the change in output voltage over the change in input voltage. If the slope has a magnitude of less than one, the output voltage will change less than the change of the input voltage. Thus, theoretically, VinLmax and VinHmin are the points where the slope equals -1 as shown on Figure 5. The limits on the output voltage are selected to allow substantial noise margins. If we look at the specification sheet of a 74C00, we see the results of this approach. With a 5 Volt supply, the noise margins are 1 Volt for both states. The output currents are

CMOS Logic

6

specified to allow a fanout of 10. Note that the input current maximum is 1 µA, some 200 times larger than the typical. At 5nA, the typical input current is negligible.

Figure 5. VTC for a CMOS inverter showing input voltage limits. We observe that an output current of only 10 µA make this logic family difficult to use when interfacing with other logic families or other circuits. It is clear that one of these gates cannot drive any of the TTL gates. Manufacturers provide some help by developing special circuits that do have much higher current drive capabilities. Also, in an attempt to increase the speed of CMOS logic, advanced families of CMOS have been developed, HC, HCT, AC, and ACT. The H stands for High speed, the A stands for Advanced, and the T stands for circuits having input specifications compatible with TTL circuits. CMOS LOGIC GATES, MOS logic gates use a resistor or MOSFET which is awlays in the active region and conducts when the output voltage is low. The strength of the CMOS logic is that there is no current in either static state, only during the transistion. When we make logic gates from CMOS we must maintain this cutoff situation in both states. This constraint requires more complexity for CMOS gates, and hence, requires more silicon area. Twoinput NAND and NOR gates are shown in Figures 6 and 7. In both cases, one pair of transistors is in series and the other pair is in parallel.

CMOS Logic

7

Figure 6. CMOS NOR Gate

Figure 7. CMOS NAND Gate

NON-ZERO LOADS, When non-CMOS loads are connected to CMOS logic gates, you can expect current to flow. Four cases are shown in Figure 8. In all cases, we will assume only logic levels are applied to the inputs of the gates. Thus, in each case, either the p-channel or n-channel transistors are cutoff. Two cases are trivial in that no current will flow (upper right and lower left). The other two cases reduce to the case discussed in the previous chapter with an n-channel MOSFET inverter with a resistor pullup or a p-channel MOSFET with a pull-down resistor. .

Figure 8. Four cases of resistor loads on CMOS gates.

CMOS Logic

8

CMOS DESIGN EXAMPLE Interface Between CMOS and TTL CMOS logic using a 5 volt power supply, normally switches between the rails; 0-5 volts. Design a CMOS inverter/interface that can drive one 7400 gate. This design requires the determination of a suitable value of k for the MOSFETs. Assume the MOSFETs are symmetric with kn = kp and both have a threshold, VTn = VTp = 2 volts. Requirements on the input for the 7400: VinLmax = 0.8 volts, VinHmin = 2.0 volts, IinLmax = 1.6 mA, IinHmax = -.040 mA. Design the transistors in the circuit below such that the noise margins are at least 0.4 volts.

Figure 9. Circuit for design example SOLUTION: OBJECTIVE: Design a CMOS inverter/interface to drive one 7400 load. PERFORMANCE REQUIREMENTS: Must meet 7400 gate input specifications: VinLmax = 0.8 v, VinHmin = 2.0 v, IinLmax = 1.6 mA, IinHmax = -0.04 mA Noise margins must be at least 0.4 volts. (VoL2.4) CONSTRAINTS: The n-channel and p-channel FETs are to be symmetric; ie. kn = kp. Threshold voltage = 2.0 volts 5 volt power supply Interface driven by other 5 volt CMOS, High = 5 v, Low = 0 volts DESIGN: The design in this case consists of deciding on the value of k for the two transistors. With the input voltage to the CMOS interface cirucit low, the output will be high, the nchannel will be off, and the p-channel will be on and supplying 40 microamps to the load gate. Depending on the output voltage, the p-channel could be either in triode of CMOS Logic

9

pinchoff. However, because this current is much lower than for the output low case, and the k-value for both transistors is to be the same, the output-low case will be the determining case. In the second case, the voltage at the input to the interface is 5 volts, the p-channel will be cutoff and the n-channel will be on. We will design the output voltage of the interface circuit to be at 0.4 volts. The n-channel will be in the triode region and must sink 1.6 mA. In = 1.6 mA = k(2(Vin - VT)Vo - Vo2) = k(2(5-2)0.4 - 0.16) = k x 2.24 or k = 0.714 mA/V2 We are now in the position to calculate the voltage when the output is high. If we assume the p-channel is in triode, Ip = 0.04 mA = k(2(VDD - VT - Vin)(VDD - Vo) - (VDD - Vo)2) = 0.714(2(5-2-0)(5-Vo) - (5 - Vo)2) or 0.714(5 - Vo)2 - 4.286(5 - Vo) + 0.04 = 0 Solving, Vo = 4.991. Clearly, the p-channel is in triode and the high output voltage easily exceeds the requirements. POWER AND ENERGY DISSIPATION During the transitions of a CMOS gate, L-H, and H-L, current flows through both transistors. This current results in a power dissipation in the devices. p(t) = i(t)v(t) Total power in the combined transistors is p(t) = i(t)VDD To use a simple example, assume a symmetric CMOS inverter where kn = kp = k and VTn = VTp = VT Also assume Vin(t) = at, a ramp function as shown in Figure 10. When Vin = VT, the nchannel turns on, and when Vin = VDD - VT, the p-channel turns off. In between, both are conducting and power is being dissipated.

CMOS Logic

10

For the first half of the transition, the n-channel is in pinchoff, the current is

Figure 10. Input voltage ramp function

Figure 11. “Square” wave input

i(t) = k(Vin(t) - VT)2 and the power dissipation is p(t) = i(t)VDD = VDD k(Vin(t) - VT)2 Energy dissipated during first half transition is E = ∫ p(t )dt

E=

Vin =Vdd / 2

∫ kV

DD

(at − VT ) 2 dt

Vin =Vt

For first half transition, the total energy dissipated is kV V E = DD DD − VT 3a 2

3

We have assumed a symmetric CMOS gate so the energy dissipated during the total transition is twice the first half, or 2 kVDD VDD ET = − VT 3a 2

3

We did the above calculations with a positive going transistion. The result would have been the same if the transistion were falling with the same slope.

CMOS Logic

11

Now let’s look at the energy dissipated when a “square” wave shown in Figure 11 is used to drive the inverter. We assume the square wave has a rise and fall time with the slope a which we used in the calculation with a single transition. The frequency is f = 1/T Average power over any given time is the energy dissipated during each transition, divided by the time between transistions. There are two transitions per period, so the average power is twice the energy per transition divided by the period. PAvg = E/T = Ef 3 4 fkVDD V DD PAvg = − VT 3a 2 Note that the power increases with frequency, k, and VDD. Power decreases with a (slope of Vin) and as VT approaches ½VDD. Example: VDD = 5 volts, k = 1 mA/V2, VT = 1 volt, Frequency = 1 kHz, a = 5V/ns ( rail-rail in 1ns ) PAvg =

4 x10 3 x1x5 3 2.5 − 1) 9 ( 3x5x10

PAvg = 2.08 µW However, if the frequency goes to 100 MHz, (Period = 10 ns) PAvg = 208 mW more than a typical IC can dissipate without getting too hot. Capacitive Loads

The power dissipation problem is made worse with capacitive loads. The capacitor charges to VDD when gate output is high, and, discharges to ground when gate output is low. The currents are shown in Figure 12, first as the output goes high, the capacitor charges toward the power supply, and then as the output goes low when the capacitor discharges to ground.

CMOS Logic

12

Figure 12. Capacitive load currents with rising and falling transitions. With each cycle (L-H, H-L), a fixed charge or energy goes from power supply to ground. E = ∫ VDD idt The total charge is Q = ∫ idt so the total energy dissipated in each cycle is E = QVDD The charge transfered is Q = CVDD, making the energy for each cycle E = CVDD2 Average power dissipated due to the capacitive load with a square wave is PAvg = fCVDD2 Again, power increases with frequency. Example: Use the same gate as previous example with 20 pF load capacitance and calculate the power dissipated due to the capacitive load. At 1 kHz PAvg = 103x20x10-12x52 = 0.5 µW If the frequency is raised to 100 MHz PAvg = 50 mW Independent of VT and k. In any system, the total power dissipation is the sum of power due to transition and load capacitance.

CMOS Logic

13

CMOS gates can be designed for reduced transitional power dissipation, but not capacitve load power loss. These losses must be controlled by limiting wiring and load input capacitance. It is interesting to note that CMOS gates inherently have a problem with input capacitance; the gate is essentially a capacitor. Input capacitance is limited by minimizing gate area.

CMOS Logic

14

EXERCISES 1.

a. For the CMOS inverter shown, label the p-channel and n-channel devices. b. Assume VT=1.5 volts. If Vin = 1.0 volts, What are the states of the two FETs? n-channel ______

p-channel ______

c. If Vin = 4 volts, what are the states of the two FETs? n-channel _____

p-channel _____

d. If Vin = 2 volts, what are the states of the two FETs? n-channel _____

p-channel _____

e. If Vin = 3 volts, what are the states of the two FETs? n-channel _____

p-channel _____

2. Determine the output voltage and the currents in the two transistors for the following circuit. kn = kp = 2 mA/V2, Vtn = Vtp = 1.5 volts.

3. Determine the total power dissipation of a CMOS gate if VDD = 5 volts, C = 25 pF, and frequency = 300 MHz, with Rail-Rail input switching transition of 0.5 ns. The CMOS transistors are symmetric with VT = 1.5 volts, k = 0.5 mA/V2 PAVG Transistion = _______

CMOS Logic

PCapicitive = ________

PTotal = _________

15

Problems

1. A CMOS inverter is to be designed such that it will drive 2 74LS00 gates. Because the largest load current occurs when the output of the inverter is low, this condition will drive the design. Determine the value of kn such that VoL = 0.5 volts and IOL = 0.8 mA with two 74LS00 loads. Then assuming kp = kn, determine VoH with 2 74LS00 loads (IinH = 0.02 mA for each 74LS00). Assume VT = 1.8 volts, power supplies of 5 volts, and the input signals to the inverter are 0 and 5 volts. 2. Using the k values from the first problem, determine the maximum current that occurs in the CMOS inverter during the switching transition. Make this calculation with no load. 3. Draw a compatibility chart showing how many loads of each kind each of the following gates can drive: 74LS00, CD4001, 74HC00, 74HCT00. Assume all are operating at 4.5 or 5 volts, whichever is appropriate. 4. A CMOS inverter gate is made with identical n and p-channel parameters, VT = 1.5 volts and k = 1 mA/V2. This gate is used to drive an LED at a curent of 20 mA with a voltage drop of 1.6 volts. The input voltages to the gate are 0 and 5 volts and the power supply voltage is 5 volts. What value of resistor should be used and what is the drainsource voltage across the drive transistor? 5. A CMOS inverter gate is made with identical n and p device parameters, VT = 1.5 volts and k = 5 mA/V2. This gate is used to drive an LED at a curent of 20 mA with a voltage drop of 1.6 volts. The input voltages to the gate are 0 and 5 volts and the power supply voltage is 5 volts. What value of resistor should be used and what is the drain-source voltage across the drive transistor? 6. A CMOS inverter gate is made with identical n and p-channel parameters, VT = 1.5 volts and k = 1 mA/V2. This gate has a capacitive load of 100 pF and is driven with a 0-5 volt square wave at 10 MHz. Calculate the power dissipation in the gate. The input square wave switches rail-rail in 10 ns. 7. A CMOS inverter gate is made with identical n and p-channel parameters, VT = 1.5 volts and k = 5 mA/V2. This gate has a capacitive load of 100 pF and is driven with a 0-5 volt square wave at 10 MHz. Calculate the power dissipation in the gate. 8. A CMOS inverter has a 5 volt power supply. The transistors are identical with k = 5 mA/V2 and VT = 1.00 volt. If the input voltage is 3.00 volts, find the output voltage. 9. A CMOS inverter has VT = 1.0 volts. The data sheet specifies that VoH = 4.3 volts with an output current of -4 mA, when Vin = 0 volts, and VCC = 5.0 volts. Determine the value of k, assuming both the n-channel and p-channel transistors are the same.

CMOS Logic

16

10. A symmetrical CMOS inverter has VT = 1.0 volts, and k = 1.00 mA/V2. Determine the average power dissipation if the input voltage has a rail-rail transition time of 1 ns and a frequency of 100 MHz. The inverter has a 100 pF load, and VCC = 5.00 volts. 11. A CMOS inverter with a 5 volt power supply is connected to an external load. With the input voltage at 5 volts, the output voltage is 1.00 volts. At this point, the inverter is sinking 2.00 mA from the external load. What is the value of k for the n-channel MOSFET? VT = 1.00 volts. 12. A CMOS Hex inverter (6 inverters on the same IC chip) is being used in the laboratory. The transistors are identical with k = 4 mA/V2 and VT = 0.5 volts. Unfortunately, the technician hooks up the 5-volt power supply but allows the inputs to float and a static voltage buildup causes the inverters to be biased into the active region. What is the maximum power that might be dissipated in the IC chip?

CMOS Logic

17

CHAPTER 3

BIPOLAR JUNCTION TRANSISTORS A bipolar junction transistor, BJT, is a single piece of silicon with two back-to-back P-N junctions. However, it cannot be made with two independent back-to-back diodes. BJTs can be made either as PNP or as NPN. The circuit symbols and representations of their configuration are given below. They have three regions and three terminals, emitter, base, and collector represented by E, B, and C respectively. The difference in the circuit symbols is the direction of the arrow. As we shall see shortly, the direction of the arrow indicates the direction of the current in the emitter when the transistor is conducting normally. As a way to remember which is which, a former student explained to me that NPN stands for "Not Pointing iN".

Figure 1. BJT symbols and representations In digital electronics applications, we are interested in using the transistor as a switch and will concentrate on switching characteristics, rather than the linear properties. We will start with an overview of the operation of the transistor. Because most bipolar switching circuits use NPN transistors, we shall concentrate primarily on them. There are four possible combinations, the base-collector junction may be either forward or reverse biased, and the base-emitter junction can also be biased either way. The four possibilities are shown in Figure 2 which also shows the operating region for each combination.

Figure 2. The four operating conditions

Bipolar Junction Transistors

1

The reverse active region is seldom used, but as we will see, it occurs in TTL gates. We begin our study of transistors by looking at each region of operation.

Figure 3. Common Base Configuration Showing Biasing Arrangement for Cutoff CUTOFF REGION (Both junction reverse biased) Let us start with the cutoff region, both junctions reverse biased as shown in Figure 3. With reverse biasing, we can assume that all currents are zero. We know that there are leakage currents associated with reverse biased junctions, but these currents are small and will be ignored.

Figure 4. Common base configuration with biasing for the forward active region FORWARD-ACTIVE REGION (BE junction forward biased, BC junction reverse biased) The biasing condition for the forward active region of operation is shown in the Figure 4. The BE junction is forward biased and the BC junction is reverse biased. In this case, the forward bias of the BE junction will cause the injection of both holes and electrons across the junction. The holes are of little consequence because the doping levels are adjusted to minimize the hole current. The electrons are the carriers of interest. The electrons are injected into the base region where they are called the minority carrier even though they greatly outnumber the holes. They "pile up" at the BE junction. From there they diffuse across the base region due to the concentration gradient. Some are lost due to holeelectron recombination, but the majority reach the BC junction. At the BC junction, the electrons encounter a potential gradient (due to the depletion region) and are swept across the junction into the collector region.

Bipolar Junction Transistors

2

Figure 5. Common Base Configuration in forward Active region Showing Internal Currents As shown in Figure 5, there are several components of current: 1. Holes injected from B-E. This is small and is ignored. 2. Electrons injected from base to emitter. This is ≈-IE 3. Electrons that reach the collector. This is -αFIE . 4. Recombination current. This is -(1 - αF)IE . 5. Collector reverse saturation current IC0 , which we will usually neglect. The term αF is the forward current transfer ratio. This term refers to the fraction of electrons that reach the collector from those that are injected into the base region across the emitter junction. In most transistors, αF is close to unity, typically 0.9-0.99. The difference between the current that is injected into the base region and that reaches the collector, becomes the base current. Thus, in most transistors, the base current is quite small, and the collector and emitter currents are close to the same magnitude. The sum of currents entering the three terminals must equal zero, IB + IC + I E = 0 (1) Also from current components 3 and 5 above, IC = -αFIE + IC0 (2) These two equations describe the currents in the common-base configuration in the forward-active region as shown in Fig. 5. We can go back to the cutoff region from here by reducing the bias voltage between base and emitter. Since the BE junction is just a P-N junction, this current is essentially a diode current. Reducing this bias voltage to zero will reduce the emitter current to zero. In this case, -IB = IC =IC0 . We will hang on to this leakage current for a while yet before we completely drop it.

Bipolar Junction Transistors

3

SATURATION REGION (Both junctions forward biased) In the active region, the collector current is proportional to the emitter current (plus the leakage current, IC0 ). This implies that the voltage bias across the base-collector junction is unimportant. But if we look again at Fig. 5, we see a resistor in series with the collector lead. If the current increases to a point that the voltage drop across the resistor plus the collector supply voltage begins to forward bias the collector junction, then holes will be injected into the collector region from the base. This hole current will counteract increases in electrons coming from the emitter, effectively limiting the transistor current. The base-collector voltage VBC at which this limiting effect begins is at about VBC =0.4 Volts, and becomes fully limiting at about 0.6 Volts. This region of operation is known as saturation. Please note that our terminology can get a little confusing. In this case, saturation refers to the circuit, not the transistor. The transistor could carry more current but the external circuit, the voltage source and resistor in the collector circuit, limits the current. REVERSE ACTIVE REGION (BE junction reverse biased, BC junction forward biased) In this case, the biasing arrangement is just the reverse of the forward active region. The collector junction is forward biased, while the emitter junction is reverse biased. The operation is just the same as the forward active region, except all voltage sources, and hence collector and emitter currents, are the reverse of the forward bias case. Another difference is that αF is replaced by αR . The equation corresponding to Eq. 2 is: IE = -αRIC + IE0 This configuration is rarely used because most transistors are doped selectively to give forward current transfer ratios very near unity, which automatically causes the reverse current transfer ratio to be very low. NOTE Please be aware that the above discussion and resulting equations are only approximations to the total operation of a transistor. A number of other phenomena are simultaneously occurring, but for typical applications, these phenomena cause only small errors in analysis. Such phenomena are: surface leakage currents, current due to holes injected from base into the emitter, hole-electron thermal generation, body resistance of the transistor, and avalanche multiplication. It is not our intention to study all effects in the transistor in this course, but to develop means to reduce the characteristics to simple, but appropriate models which will allow us to easily analyze transistor switching circuits. However, in the process of developing and using a model, an engineer must develop a feeling for when it is appropriate to use such a model and the limitations on the model.

Bipolar Junction Transistors

4

COMMON BASE AMPLIFIER Before we go on to the common emitter configuration, we should take a few moments to observe how a transistor can be used as an amplifier. Again referring to Fig. 5, assume that VEE is increased by 1mV. This change in voltage would cause an increase in the current in the PN junction between the base and emitter. Most of this current would be reflected in an increase in the collector current. This change in collector current would manifest itself in a change in the voltage across the collector resistor, RC . Now if the collector resistor is much larger than the emitter resistor, we would see a large voltage gain. We will not pursue this path further as this topic is the subject of another course.

a. Common Base b. Common Emitter c. Common Collector Figure 6. Basic Configurations and Biasing for the NPN Transistors COMMON EMITTER CONFIGURATION The common-emitter configuration is much more prevalent than the common base configuration, especially in digital or switching circuits. This configuration is shown in Figure 6 along with the common base and common collector configurations for comparison. The name comes from the fact that the emitter terminal is held at ground or "common". If loop equations are written for both the input and the output circuits, the two currents of interest will be the collector current, IC, and base current, IB. Equation 2, above, is written in terms of the emitter and collector currents. By substituting Equation 1 into Equation 2, we can solve for the collector current in terms of the base current,

Where we have dropped the subscript on α, since we are assuming forward active region. It is convenient to define a new variable,

Thus, Equation 3 becomes,

The terms β or hfe relate a change in base current to a change in collector current, or current gain. If we ignore ICO, then the dc current gain, hFE, is the ratio of collector to base currents. β is used interchangeably for both ac and dc current gain. Many times we use β and hFE interchangeably.

Bipolar Junction Transistors

5

CIRCUIT MODELS FOR A COMMON EMITTER TRANSISTOR. BASE-EMITTER CIRCUIT When we go back and look at Figure 6, we can see that the relationship between base current and base-emitter voltage is controlled by the I-V characteristic of the base-emitter PN junction. While there is some effect caused by the collector-emitter voltage, these effects are quite small and for the purposes of this course, will be neglected. (The course in linear electronics does not neglect this effect, which is often expressed in terms of rµ.) Thus, we can consider the base-emitter junction to operate just like a diode, and model it similarly. When talking about digital logic gates, we will assume that there will be no base current if the base-emitter voltage VBE is less than VBEγ = 0.50 Volts. We will also assume that if the base is driven heavily (into saturation), the base-emitter voltage VBE is equal to VBEsat = 0.80 Volts. Sometimes it is convenient to use an intermediate value of 0.7 volts when the transistor is in the active region. We will also use voltages of 50 mV higher for the ECL gates which are discussed later. Figure 4 shows the circuit models we will use for the three regions of operation for the NPN transistor in the common emitter configuration. The base-emitter voltages are shown as discussed. Next we will. develop the collector models.

Figure 7. Common Emitter Circuit Models for an NPN Transistor COLLECTOR EMITTER CIRCUIT MODELS The collector emitter circuit is more complex to model. Again we will talk about the three regions of operation. The collector characteristic is a plot of IC vs VCE for a few representative values of base current. Typical collector characteristics are given in Figures 8 and 9; the forward characteristic in Figure 8 and the reverse characteristic in Figure 9. Because the reverse active region is seldom used, we will not develop that model here.

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Figure 8. Forward collector characteristics for a 2N3903

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Figure 9. Reverse characteristics for a 2N3903

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Cutoff If the base-emitter voltage is less than cut-in ,VBEγ, the base current is zero. From Equation 3, we see that the only current is the leakage current which we usually consider negligible. Thus, we can consider the transistor in cutoff to be an open circuit between the three terminals as shown in Figure 7a. The limit is, of course, that VBE < 0.50 Volts. Active Region The discussion of the active and saturation regions rests on Equation 5 and the collector characteristic graph which is shown in Figure 8. The cutoff region is shown on the collector characteristic as the single line where the base current is zero; in this case the xaxis or IC =0. The active region is the large middle part of the family of curves where the collector current is essentially constant for a fixed base current. The saturation region is the left-hand portion of the curves where the curves are almost vertical. The curves in the active region correspond closely to Equation 5. The current is only a function of base current, not a function of collector-emitter voltage, VCE . In this case, a suitable model of the transistor is shown in Figure 7b. This figure shows the base-emitter model as a voltage source and the collector-emitter model as a current-controlled current source, βIB . The defining limitations are IB >0 and VCE >0.2 V. Again, you should be aware that this model is only an approximation to the real device. For instance, collector current does vary slightly with VCE and the spacing between the curves is not exactly constant; β varies somewhat with collector current. The course in linear electronics more closely models the first of these effects with an additional parameter, ro . Saturation Region The saturation region model is perhaps the poorest approximation to the real device. The saturation region is modeled simply as a fixed voltage of VCE = 0.2 Volts. In this region of operation, the base circuit is modeled as VBE = 0.80 Volts. This model is shown in Figure 4c. The limitations are that IB >0 and IC VCEsat (=0.2V), this result is consistent with the assumption that the transistor is operating in the active region. Now let us look at the final case. Vin = 10 Volts If we assume the transistor will be in the active region, the process will be the same as for the previous case. We can look at Figure 13 but with the input voltage at 10 volts.

Proceeding to the collector circuit, we will attempt to find the collector current as we did before; IC = βIB = 20 * 1.86 = 37.2 mA The output voltage is then,

Bipolar Junction Transistors

(6)

12

VO = VCC - I R = 10 - 37.2 * .500 = - 8.6 Volts ??

(7)

This result says that the output voltage is negative. How can that possibly be? There is no source for the negative voltage; no negative power supply. The transistor is modeled as a current-controlled current source, but is not a current generator. It can only work within the limits of the power supplies. You will note on the collector characteristic curves that the collector voltage cannot go negative with a positive collector current. The obvious conclusion is that the transistor circuit has saturated, and the active region model is no longer valid.

Figure 14. Circuit in Saturation If we go to the saturation region model, the circuit is shown in Figure 14. Note that the only base circuit change is to change VBE to 0.80 volts. This change makes the base current

We can also determine the collector current,

We already know that the output voltage is 0.2 Volts (=VCEsat) since the transistor is in saturation. The only thing left is to verify that the model used is appropriate for the situation. (We already know the active region model is inappropriate, and that we are not in cutoff, IB >0.) To demonstrate that the saturation model is appropriate, we need only show the ICsat 19.6, the saturation model is appropriate. SUMMARY It is appropriate at this time to explicitly define the criteria for operation in each mode. Cutoff: VBE < VBEγ and VBC < VBCγ If this condition occurs, the base current will be zero and the collector current will be zero. Of course, we are ignoring the leakage currents. We are also not considering the possibility of using the transistor backwards with collector and emitter reversed. Thus, we are assuming that the collector voltage is more positive than the base voltage. We will discuss the value of VBEγ later. Active Region IB > 0, VBE = 0.70, IC = βIB , VCE > 0.2 V.

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Saturation Region: IB > 0, VBE = 0.80, VCE = 0.2, IC < βIB . The value of VBEγ we choose will have a significant bearing on how we view the transition between cutoff and the active region. For logic systems, we will use VBEγ = 0.50 Volts. This is the value the base-emitter voltage must be less than to guarantee the transistor is cutoff. Note that when the transistor is saturated, VBE =0.80. For logic systems, any value in between is indeterminate. This inconsistency will not cause us any trouble in logic systems. However, this discrepancy will cause some difficulty when we are trying to find the transition between cutoff and active operation and between active and saturation operation. For our next example, we will discuss a transistor as an inverting amplifier that varies continuously between cutoff and saturation. INVERTING AMPLIFIER To get a better feel for how a transistor works with real signals, we will discuss a transistor inverter with a sine wave input. The transistor will operate in all three modes. The circuit is shown in Figure 15. Temporarily, we will assume VBEγ = 0.75 = VBEact = VBEsat In other words, there is no discontinuity from cutoff to the active region, and no discontinuity from the active region to saturation. Where do we start the analysis of such a circuit? First let us consider the possibilities. At any given instant of time, the transistor will either be cutoff, in active region, or saturated. From our previous example, we should note that if the input voltage is below a certain value, the transistor will be cutoff; above that value, the transistor will be in the active region with positive base current, and with the input voltage higher still, we get enough base current to saturate the transistor. Our job here is to determine which region or regions are utilized. First, let us look to see if the transistor is cutoff. Model: VBEγ = VBEactive = VBEsat =0.75 VCEsat = 0.2 Figure 15. Transistor Inverting Amplifier

β = 40

Cutoff

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In order for the transistor to conduct at all, the input voltage must be above VBEγ. Conversely, if the input voltage is below VBEγ, the transistor will be cutoff. Does that occur here? Yes, the input voltage may go as low as -5 Volts. Thus, for all the time the input voltage is below VBE =0.75 Volts, the transistor is cutoff. The transistor is replaced by its cutoff model in Figure 16. What then is the output voltage during this time? Since there is zero collector current, zero current through the collector resistor, there is zero voltage drop across the resistor, and the output voltage is VO = VCC = 10 Volts. (10) Figure 16 shows the circuit with the transistor replaced with its cutoff model and Figure 17 shows the input and output waveforms during the cutoff period.

Figure 16. Circuit With Transistor Cutoff

Figure 17. Vo When Transistor is cutoff

Active Region

Figure 18. Transistor Inverter Circuit During Active Region If the input voltage goes above 0.75 V, the transistor goes into the active region. If we use the active model for the transistor and place it into the circuit as shown in Figure 18, we can then analyze the circuit. The base current is

Thus, the base current will have a sinusoidal component like the input voltage, only shifted. Note that this equation is valid only if Vin >VBEγ. The next question, of course,

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is what is the output voltage? Since the collector current is βIB = 40IB , the output voltage is the supply voltage minus the drop across the collector resistor.

Again, the output voltage has a sinusoidal component. This equation is valid only for Vin >0.75 Volts, and also as long as the circuit is not saturated, Vo = VCE >0.2 v. Thus, the output voltage is sinusoidal only for a portion of the cycle. It should also be noted that if the input voltage = 0.75 Volts, the output is equal to 10 Volts, which is consistent with the cutoff case, no discontinuity. We now need to determine if the circuit saturates. Saturation We note from our earlier example that the circuit will be in saturation if the collectoremitter voltage drops to 0.2 Volts. We could solve Equation 12 to find what input voltage would cause this to happen. However, we will take a more straightforward approach. Let us simply look at the saturation model for the circuit to determine what the input voltage must be in order to reach saturation.

Figure 19. Transistor Inverter Circuit in Saturation The saturation model for the circuit is shown in Figure 19. We now work backwards from the collector circuit. In Figure 19, the collector current is

The minimum base current required to support this collector current is

The minimum input voltage required to furnish this much base current is

Vinsatmin = VBEsat + IBsat* RB = 0.75 + 0.245 mA *10 KΩ = 3.2 Volts.

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(15)

16

Since the input voltage rises above this value, the circuit does indeed saturate and will be in saturation whenever Vin>3.2 volts. In saturation, the output voltage is 0.2 Volts. Thus, Figure 20 shows the output voltage for the entire cycle. Figure 20 shows that the circuit spends relatively little time in the active region. However, during this time the output voltage has a sinusoidal component and that sinusoidal part has a negative sign with respect to the input voltage, hence, the name of inverter. Actually this circuit is not really suitable for a linear signal inverter without substantial modifications. The linear inverting amplifier is discussed in more detail in the linear electronics course. It is useful to note, however, that the circuit we have analyzed does "square up" a sine wave, making the signal useful for digital switching.

Figure 20. Waveforms for a complete cycle for the inverting amplifier circuit.

Voltage Transfer Characteristic It is often useful to plot the relationship between the output voltage and the input voltage, called the voltage transfer characteristic or VTC. For the previous example, we already know the transistor is cutoff for Vin < 0.75 making VOCutoff = 10 and the transistor is in the active region when 0.75 < Vin < 3.2. We also know the transistor is in saturation when Vin > 3.2 making the VOsat = 0.2. In active region: VOact = 10 - 4(Vin - 0.75) = 13 - 4Vin which is an equation for a straight line. The voltage transfer characteristic curve is shown in Figure 21.

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Figure 21. Voltage transfer characteristic for the transistor inverter

PNP Transistors The PNP transistor works the same as the NPN, except that the carriers are holes instead of electrons, and all the currents and voltages are in the opposite direction.

Figure 22. PNP transistor with internal currents. Figure 22 is a drawing of a PNP transistor showing the internal and external currents. The internal currents correspond to the same currents as for the NPN transistor shown earlier. The currents of importance are numbered 2, 3, and 4, hole currents, injected from emitter to base(#2), the diffusion current that reaches the collector (#3), and the holes lost by recombination in the base region (#4). Currents #1, and #5, are parasitic and usually ignored. Because the external currents are considered positive going into a terminal by convention, IE = current #2 IC = - current #3 = -αIE IB = - current #4 = -(1-α)IE

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The above equations relate the base and collector curents to the emitter currents as is usual for the common base configuration. However, because we usually use the transistors in the common emitter configuration, it is more convenient to reperesent the currents in terms of the base curents. We still get IC = βIB but now, IB is a negative quantity. We also still have IC + IB + IE = 0 making IE = IB + IC Now however, IE will be a positive quantity, and IB and IC are negative quantities. The currents in the PNP transistor are in the opposite direction as for the NPN transistor, making the voltage drops across the junctions in the opposite direction. Thus, for the PNP transistor operating in the active or saturated regions,VBE and VCE are negative values. A typical common emitter configuration is given in Figure 23. From the circuit model, IB = -1.3/5K = 0.26 mA The collector current is (β = 10) IC = -0.26β = -2.6 mA and the voltage at the collector is VO = -5 + 1K x 2.6 mA = -2.4 volts.

Figure 23. Typical common emitter application for a PNP transistor with its active model. The difficulty in dealing with the PNP transistor is keeping track of the negative currents. Because most logic systems use positive voltages and because it is more intuitive to have currents going downhill instead of going uphill as in the above example, circuits can be drawn as shown in the application in Figure 24 as an output interface in a logic system. .

Figure 24. PNP application in a logic system interface

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In this case, we will assume the logic gate output is either +5 volts, or 0 volts. (Later on, we will use more accurate data for the gate.) If the gate output is high at +5, the voltage across the base-emitter junction is zero volts, which will keep the transistor cutoff. The circuit with its cutoff model is shown in Figure 25. Note that we specify the emitter-base voltage as VEB. In this case, VEB = 0, because both terminals are connected to +5 volts.

Figure 25. Cutoff model

Figure 26. Saturation model

If the gate output is low, or zero volts, the emitter-base junction is turned on. We suspect the transistor is saturated, so we use the saturation model in Figure 26 for the analysis. It is convenient to use positive currents I1 (= - IB) and I2 ( = - IC) instead of their negative counterparts. Note again, we use reverse polarities for the the two terminal voltages. We can calculate the two currents I1 = (5-0.8)/5K = 0.84 mA I2 = (5-0.2)/1K = 4.8 mA If β = 10, the transistor is obviously in saturation, and VO = 4.8 volts. Notice that the voltage across the 1K load resistor is 4.8 volts.

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EXERCISES 1. For the following circuit, assume the following model for the transistor: VBEγ = 0.5 v, VBEact = 0.7 v, VBEsat = 0.8 v, β = 30, VCEsat = 0.2 v.

Circuit Diagram

Voltage Transfer Characteristic

If Vin = 5.0 volts, a. Draw arrows showing the actual current direction in each terminal of the transistor. b. Determine three terminal currents: IB = ________ IC = ________ IE = ________ c. Determine Vo. Vo = ___________ d. Find minimumVinsat and maximum Vincutoff. We can use these points to draw the voltage transfer characteristic, but because we are using a discontinuous transistor model, we cannot precisely predict the region in between. Simply draw a dotted line between the two points. 2. A ramp voltage is applied to the inverter circuit shown. A continuous model for the transistor is: VBEγ = VBEact = VBEsat = 0.7 volts VCEsat = 0.2 volts

β = 25

Plot the output voltage waveform and determine the time it takes the output voltage to go from cutoff to saturation.

3. A PNP transistor circuit is shown below. The transistor model is: Bipolar Junction Transistors

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VBEγ = VBEact = VBEsat = -0.70,

VCEsat = -0.20, β = 20

If Vin = 5.0 volts, a. Draw arrows showing the actual current direction in each terminal of the transistor. b. Determine three terminal currents: IB = ________ IC = ________ IE = ________ c. Determine Vo. Vo = ___________ d. Find Vinsat and Vincutoff. Vinsat = _________

Vincutoff = ____________

e. Draw the voltage transfer characteristic.

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DESIGN EXAMPLE: TRANSISTOR SWITCH To illustrate the utility of a transistor as a switch, let's look at the following example. We have a signal from a 74HC00 CMOS logic gate that is supposed to turn on an LED when the output of the gate is a high. The specifications of interest are: 74HC00: VOHmin = 3.84 volts when IOH = -4.00 mA, VOL = 0 at IOL = 0 LED:

VD= 1.80 volts when ID =30 mA, Imax= 60 mA

Because we want the LED current to be about 30 mA, the logic gate clearly will not supply the necessary current. A BJT switch is a good candidate for this job. A suggested circuit is shown below. In this case, when the output of the gate is high, we want the transistor to saturate with 30 mA through the diode. When the output of the gate is low, the transistor will be turned off. The high output case is shown.

Figure 27. Circuit to be designed First, let's look at the case when the gate output is low. In this case, the output voltage is zero and the transistor will be cutoff. There will also be no diode current. When the gate output is high as shown in the circuit above, we want the transistor to be saturated. If the diode current is 30 mA, and the diode voltage is 1.80 volts, the voltage across the series (current limiting) resistor, Rc, is 3.0 volts. Thus, this resistor value must be 100 Ω. Assuming the transistor βsat =10, the base current must be greater than 3 mA to guarantee saturation. In this case, the gate can source 4 mA when the output voltage drops as low as 3.84 volts. If the base-emitter voltage is 0.8 volts, the voltage across the resistor must be 3.04 volts. At a base current of 4 mA, the resistor value is 760 Ω. Note that the base current in saturation needs only to be 3 mA or greater. Thus, the base resistor can be made larger, up to 1013 Ω. The standard 5% values between these limits are 820, 910, and 1000 Ω. Any of these values is satisfactory if we consider only nominal calculations. (If we must guarantee operation with 5% resistors, 800 < RNOM < 964.) With this design, the LED will be off when the gate output is low and the LED will be on when the gate output is high. Bipolar Junction Transistors

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Problems 1. In the laboratory, take a set of transistor characteristic curves for a transistor in the forward active region. From these curves, develop the approximate, straight-line models for the three regions of operation. Present your results by drawing the circuit models with component values and by drawing the characteristic "curves" represented by these circuit models. Turn in the original collector characteristic curves, the circuit models, and the “curves” represented by these models. 2. For the circuit below, determine the values of the input voltages that will cause the transistor to be cutoff and saturated. Use the following model component values: β=50, VBEγ =0.50, VBEsat = 0.80, VCEsat = 0.20.

Problem 2

Problems 3 and 4

For Problems 3 and 4, use the following model for the transistor: β = 30, VBEγ = VBEact = VBEsat = 0.70, VCEsat = 0.20 3. Draw the voltage transfer characteristic (Vo vs Vin) for the circuit given. Let Vin go high enough for the circuit to saturate. 4. Let the input voltage be a 5 volt sine wave (5sinωt). Sketch the output waveform showing all breakpoints in the waveform and the input voltage at each breakpoint. Determine the time it takes the output waveform go from one "rail" to the other if the frequency is 60 Hz. 5. Use a computer simulation to verify the results from Problem #4. Use the library model for a 2N3903 with Beta = 30. Note that the results may not be quite the same because the PSPICE library model is different from the 3-region model we use in class and that model includes capacitance. Be sure to show on your plots the time it takes to make the transition from cutoff to saturation.

6. Find the input voltage required to cause the pnp transistor circuit to saturate.

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PNP Transistor Model:

VEBsat = 0.8,

VECsat = 0.2,

Betasat = 20

Problem 6 7. The input voltage waveform is shown. Draw the output voltage waveform showing the input and output voltages at each breakpoint. The transistor has the following model parameters: VBEcutin = VBEact = VBEsat = 0.7 volts, VCEsat = 0.2 volts, Beta = 25

8. A PNP transistor is used in the following circuit to drive a load as shown. Determine the value of the base resistor needed to guarantee saturation if the input voltage switches between +12 volts and 0.4 volts. Note that in one state, the pnp transistor will be cutoff and in the other state, the pnp transistor is to be saturated. The pnp transistor has the following model parameters: VEBcutin = 0.5 volts, VEBsat = 0.8 volts, VECsat = 0.2 volts, Beta = 20

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Problem 8. 9. A Darlington connection is shown below. If 0.1 mA is going into the base of the first transistor, how much current can go through the resistor? Beta is 20 for each transistor. What is VCE for the output transistor?

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Chapter 4

Resistor-Transistor Logic Resistor-transistor logic, RTL, is an old technology that is no longer commercially available. It is a useful starting place to study logic gates because it is a logical extension of the transistor inverter studied in the previous chapter, and because it is a useful circuit in itself. A two-input gate is shown in Figure 1. As with most logic, the transistors operate only in saturation or cutoff. There are four possible combinations of logic inputs, L-L, L-H, H-L, and H-H for inputs A and B respectively. A low input will cutoff the associated transistor, and a high input will saturate it. Figure 2 shows these combinations where the transistors have been replaced by their corresponding models. If both transistors are cutoff, the output will be high, 3.0 Volts. If either transistor is saturated, the output will be 0.2 Volts, or low. The truth table for this gate is given in Figure 3 and shows that the gate performs a logic NOR function.

Figure 1. RTL gate

Figure 3. Truth table for RTL gate

a. L-L inputs

b. L-H inputs

c. H-L inputs

d. H-H inputs

Figure 2. RTL gate with four combinations of inputs

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Our job is to determine the eight terminal specifications, input and output voltages and currents when these terminals are both high and low as shown in Table 1. Definitions of these terms are given in Appendix A. Table 1. The Eight Terminal Specifications to be Determined. VinH

VinL

VoH

VoL

I inH IoH

IinL IoL

We begin the calculation of the terminal specification by noting that when a transistor is saturated, the collector-emitter voltage is 0.2 volts. The output terminal is connected directly to the collector terminal, thus, VoL = 0.2 volts. Assume the gate on which this analysis is taking place is embedded in a logic system. Its inputs are driven by other identical logic gates. Then if the outputs of those driver gates are logic low, the inputs to the device under analysis are low. We will use this as a starting place for the calculations. VinL = 0.2, (LOW input to both transistors) Under this condition, both transistors will be cutoff, making the output high. The circuit diagram with circuit models for the cutoff transistors is shown in Figure 4. The output voltage then is 3.0 Volts. We will call this a no-load condition when there is no load connected to the output. VoH = 3.0 Volts (No-Load)

Figure 4. RTL gate with both inputs low. The transistors are replaced by the cutoff model. Also note from Figure 4 that the input current is zero because the transistor is cutoff. Thus, IinL = 0 The question naturally arises about how high the input voltage may rise and still keep the transistor cut off. The input voltage must stay below VBEγ to guarantee the transistor is off, making Resistor-Transistor Logic

2

VinLmax = 0.5 Volts. The maximum value is given in this case because we need to know how high it may rise. This means that the input voltage may rise as high as 0.5 volts and still keep the transistors off, and will always be recognized by the gate as a "low". INPUT HIGH In this case we want the transistor to be saturated. Either transistor saturated will cause the output to go low, to 0.2 volts. This case is shown in Figure 5. We show the circuit with one of the transistors cutoff, although both saturated would produce the same result. We start by determining the minimum input current that will keep the transistor in saturation.

Figure 5. RTL gate with one input high.

3.0 − 0.2 = 4.375mA 0.640k The minimum base current to keep the transistor saturated is (assuming β=30) I Csat =

I inH min =

I Csat 4.375 = = 0146 . mA β 30

The minimum value is of interest because this current is dependent on the input voltage. From this value, we can determine the minimum input voltage that will be guaranteed to be recognized as a "high". VinHmin = 0.8 + 0.146 mA x 0.450 KΩ = 0.866 Volts Any input voltage greater than or equal to 0.866 volts will cause the transistor to be saturated, and thus be recognized as a "high". We previously found VinLmax =0.5 volts. Any input voltage between 0.5 and 0.866 is an invalid logic level and the manufacturer assumes no responsibility if you provide an input voltage in that range. For the gate with no load, the relationships between input and output voltages are presented graphically in Figure 6. The noise margins NMH and NML are defined in Appendix A.

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Figure 6. Graphical representation of input voltage ranges and output voltages for the unloaded RTL gate.

CALCULATIONS WHEN THE GATE IS USED TO DRIVE OTHER GATES Up to this point we have done all calculations assuming no load on the gate. While convenient for visualization, it is not a very useful situation. Let us look at the loading of the gate as if it were embedded in a logic system driving identical gates as shown in Fig. 7.

Figure 7. Driver gate being analyzed driving load gates.

Figure 8. Driver with loads when driver output is low.

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There are two cases, the driver output being low as shown in Figure 8, and the driver output being high as shown in Figure 9. When the output is low, the driver transistor is saturated and the output is 0.2 Volts. All the input transistors of the load gates are cutoff and no current flows. The output voltage remains at 0.2 volts, unchanged by the loads.

Figure 9. RTL gate with N loads when driver output is high. However, when the output is high, the load gates require current, pulling the output voltage down. This situation is shown in Figure 9 with an unspecified number of loads. How many can you connect and still expect the gates to work right? This is called the maximum fanout. The output voltage is determined by the load current being SOURCED by the driver. VoH = 3.0 - Is x 0.64 kΩ Where the current coming out of the driver gate is divided between the load gate inputs. Since the minimum voltage at the load gate inputs when high is 0.866 Volts, this is the lowest the output voltage of the driver gate may go. Thus, I oH max = −

3.0 − 0.866 = −3.334mA 0.64 k

The negative sign comes about because currents are always defined as positive when going into the terminal. Each load requires a minimum of 0.146 mA as input current, therefore, the maximum fanout is MaxFanout ≤

3.334 = 22.8 0146 .

Thus, the maximum fanout is 22.

PRACTICAL FANOUT

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5

The maximum fanout will not provide for any noise margin. Also, manufacturing tolerances will not allow such close calculations. Therefore, most manufacturers limit the allowed fanout to a much smaller number than found by nominal and typical calculations. Typically, the fanout would be limited to say 5 loads. In this case, the output voltage under maximum load would be substantially higher than calculated above. With five loads, the output current will be I oH = −

3.0 − 0.8 = −3.014mA 0.45 0.64 + 5

and the output voltage will be VoH = 3.0 - IoH x Rc = 3.0 - 3.014 mA x .64 kΩ = 1.071 Volts Under this condition, the Noise Margin High is NMH = 1.071 - 0.866 = 0.205 Volts Note that in many cases the output voltage will be higher than the value just calculated and thus the noise margin larger. If we define the maximum allowed fanout to be five, then VoHmin = 1.071 Volts.

CALCULATION OF IoL If we define the maximum allowed fanout to be five, then the minimum voltage ever seen by the input will be 1.071 Volts. The minimum current into the input will be 1071 − 0.8 . = 0.602mA 0.45k The maximum collector current is β times this or 18.067 mA and still stay in saturation. Thus, the maximum output current when the output is low is this current minus the current down through the 640 Ω resistor. I inH min =

IoLmax = 18.067 - 4.375 = 13.692 mA We now have all the specifications for this gate. The results are shown in Table 2. We have assumed here that the fanout is limited to 5 loads. There is a certain amount of arbitrariness in the choice of VoH and IoH. Many other performance requirements could have been chosen. The values given are self consistent, however. First note that VoHmin>VinHmin and VoLmax VoHmin)

=0

= 0.2 Volts

IinHmin = 0.602 mA IoHmax = - 3.014 mA ---------------------------------------------------------------------------INTERFACING TO RTL Table 2 is a set of specifications that would be provided by the manufacturer of the RTL logic. From this information, you would be expected to be able to use the gates without exceeding any specification. Any gate embedded in a logic system using only the RTL gates analyzed, would be guaranteed to work within the specifications. The only limitation we must follow is limiting the fanout to five loads. However, logic is rarely used without interfacing to the outside world at some point. There must be some signals at some point that are generated outside the logic system that must be recognized as a logic one or a zero. Similarly, there is usually some point where a logic gate output must drive a device that is not another gate.

a. b. c. Figure 11. Three possible configurations of a switch-to-logic gate interface.

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7

Figure 11 shows three possible connection using a switch to generate the logic inputs to a gate. In Figure 11a the switch is used as a pull-down to make the input voltage a logic zero when the switch is closed. The resistor pulls the input voltage up to a logic one when the switch is open. Looking at the specifications in Table 2, we note that when the switch is closed, the input voltage is zero volts, well below the required VinLmax of 0.5 volts. The gate input current is zero so the switch needs only to handle the current coming down through the resistor. When the switch is open, current will come down through the resistor into the gate input terminal. The specifications show the voltage must be above 0.866 volts and the current must be at least 0.602 mA. From these values, we can select a value of R1. To provide a noise margin, we should certainly provide a voltage at least as high as VoHmin = 1.071 volts. Let's shoot for 2.0 volts to provide even greater noise margin. We know this will provide greater than the minimum current, but we don't yet know how much. We now have to put on our engineer's hat and look inside the gate to finish the design. The parts of the circuit that pertain to this case are shown in Figure 12 with only the base-emitter voltage of the transistor shown. The transistor must be in saturation so we use 0.8 volts for VBEsat.

Figure 12. Circuit with switch open. Solving this circuit, we can write a single node equation at the input node, Vin = 2.0 v, (3.0-2.0)/R1 = (2.0-0.8)/450 Solving, we get R1 = 375 Ω. This is not a standard resistor value so we would select a nearby standard value and re-solve the circuit, this time to determine the input voltage. If we are satisfied with the result, then we are finished. If the input voltage found is not satisfactory, we would try with another value. Note that we do not have to get 2.0 volts, we simply chose that value as a place to start the design. We do, however, have to provide Vin > VinHmin. The design of the circuit in Figure 11b would follow a similar track. Notice that in this circuit the input voltage is high when the switch is closed; the opposite of the circuit in Figure 11a. Figure 11c is simpler than 11b, but it is not recommended because when the switch is open, the input lead of the gate is floating or disconnected and susceptible to noise which might cause random errors in the logic.

Resistor-Transistor Logic

8

Figure 13 shows an example of an interface at the output of the gate. In this case, we want to drive an LED at 30 mA. Neither the high nor low output currents of the gate are sufficient, so we use a transistor as a current amplifier. Many other configurations are possible and some may be preferrable depending on the application. We will not go through the design of this circuit here. A similar circuit was designed at the end of the previous chapter.

Figure 13. Example of an output interface circuit.

Resistor-Transistor Logic

9

EXERCISE Mountain State Electronics, Inc. has provided a data sheet for its line of RTL NOR gates: VinLmax = 0.45 volts

VoLmax = 0.3 volts

VinHmin = 0.85 volts

VoHmin = 1.00 volts (@N=5)

IinL

IoLmax = 10.00 mA (@ Vin > VoHmin)

= 0.0 mA

IinHmin = 1.00 mA 1. Determine:

IoHmax = -5.50mA (@Vo = VoHmin)

Fanout = _______ NMH = _______ NML = _______

2. For the logic network below, write the logic level ( H or L) at each node. 3. If you used a voltmeter to measure the voltage, what voltage would you expect to measure at each node? (Use signs to describe the voltages. These voltages will be the thresholds you will use to determine if the network is operating properly.) VA ______ VC ______ VE ______ VG ______ VB ______ VD ______ VF ______ VH ______ 4. Beside each connecting wire, draw an arrow showing the direction of the actual current in the wire. Write I = 0 if no current flows. 5. In each wire labeled A-H, estimate the magnitude of the current. IA ______ IC ______ IE ______ IG ______ IB ______ ID ______ IF ______ IH ______ 6. Detemine the maximum allowable value of the pull-up resistor R at the input of gate #1. Rmax = _________

Resistor-Transistor Logic

10

In-Class Design Exercise: Assume RTL Specifications are given by manufacturer: Terminal Specifications: VinLmax = 0.5 volts

IinL = 0

VinHmin = 0.866 volts

IinHmin = 0.602 mA For max IoL

VoL = 0.2

IoLmax = 13.692 mA @VinH>VoHmin)

VoH = 1.071 @Fanout = 5

IoHmax = -3.014 mA

1. Drive an LED from an RTL gate.

Method 1

Method 2

LED Model: VLED = 1.8 for I > 0 I = 0 for VLED < 1.8 Performance Requirement: 9 mA < ILEDon Constraints: Use 5% standard value resistors, nominal calculations 3-volt power supply Method 1

How does it work? (Draw the currents on the diagram)

When gate output is low LED is _________ When gate output is high LED is _________ How much current is available to turn on the LED? ______ R1 = ______

Resistor-Transistor Logic

11

Method 2

How does it work? (Draw the currents on the diagram)

When gate output is low LED is _________ When gate output is high LED is _________ How much current is available to turn on the LED? ______ ________< R2 < ________

2. Same problem, except the performance requirement: 20 mA < ILED < 25 mA Sssuggested Mthods:

Method 3 Method 4 Assume the following transistor specifications: VBEγ = 0.5, VBEsat = 0.8, VCEsat = 0.2, β = 10 (negative voltages for pnp) Method 3

How does it work? (Draw the currents on the diagram)

When gate output is low, the transistor is ______, and LED is _________ When gate output is high, the transistor is _____, and LED is _________ How much current is available to saturate the transistor _______ to turn on the LED? ______ R3min = ________,

______< R4< ______

Designer’s choice: R4 = ______,

then ICsat = _______, IBmin = _______

R3max = ________ Resistor-Transistor Logic

12

Method 4

How does it work? (Draw the currents on the diagram)

When gate output is low, transistor is ______, and LED is _________ When gate output is high , transistor is _____, and LED is _________ How much base current is available to saturate the transistor _______ What is the maximum collector current is available to turn on the LED? ______ R5min = ________,

______< R6< ______

Designer’s choice: R6 = ______,

then ICsat = _______, IBmin = _______

R5max = ________

Problems Mountaineer Logic Inc. is developing a new line of RTL logic NOR gates. The gates are said to have a fanout of 5 and the following component values: VCC = 5.0 Volts, RC = 470 Ohms, RB = 1.5 K Ohms, and βmin = 20. 1. For the RTL gate above, present the following specifications in tabular form: Noise margins NMH and NML, VoL, VoH (at N=5), VinLmax, and VinHmin. 2. Draw the voltage transfer characteristic (Vo vs Vin ) for the gate. 3. Determine the absolute maximum fanout with zero noise margin.

4. The RTL gate shown below has a specified “high” noise margin, NMH, of 1.00 volts. What is the specified fanout? The transistors model parameters are: Resistor-Transistor Logic

13

VBEcutin = 0.5 volts, VBEsat = 0.8 volts, VCEsat = 0.2 volts, Beta = 12

5. A logic family has the following terminal specifications: VinLmax = 0.6 VoLmax = 0.4 VinHmin = 1.8 VoHmin = 2.4 IinL = -1.0 mA IoLMax = 10 mA IinH = 1.0 mA IoHmax = -5 mA This family of logic gates is used in the following logic system.

What voltage do you expect at nodes A and B? What values of resistors RC and RD can be used? (Specify max or min.)

Resistor-Transistor Logic

14

Chapter 5

DIODE TRANSISTOR LOGIC Resistor-Transistor Logic was an early form of logic used in the 1950's and early 1960's. RTL was made from discrete transistors and resistors and manufactured on printed circuit boards with several gates per board. These boards were plugged into board sockets with wiring on the socket pins determining the system function. RTL was a big improvement over vacuum tube technology previously used, requiring less than one quarter the space and one tenth the power dissipation. RTL was superceded in the 1960's by DiodeTransistor Logic and then Transistor-Transistor Logic, DTL and TTL respectively. DTL was initially made with discrete transistors and resistors before being integrated onto silicon. One early form of DTL, used by IBM Corp in the 360 family of computers, was really a hybrid technology. Transistor and diode chips were glued to a ceramic substrate and aluminum resistor paste was deposited on the substrate to make resistors. Finally the ceramic base and components were hermetically sealed in an aluminum can. This family was used extensively in IBM products in the middle to late 1960's. While this family was not a true integrated circuit, it was very successful and was less expensive than true integrated circuits for several years. By the early 1970's integrated circuits became quite common and DTL gave way to TTL which was more appropriate to integrated circuit technology. While DTL is no longer commercially used, we will discuss it because it is similar to and easier to understand than TTL, and because designers still find the configuration of value. First, however, we will discuss diode logic which is the front end of the DTL gate and performs the actual logic operation. DIODE LOGIC

Figure 1. Diode logic and truth tables Two diode logic configurations are shown in Figure 1. The truth tables show that the first circuit performes a logical OR and the second circuit performs a logical AND. One problem with these circuits is that there is a voltage level shift through the circuit. If several circuits are cascaded as shown in Figure 2, the output voltage for each stage is approximately one diode voltage drop further away from the rail. Logic "0" rises for the Diode-Transistor Logic

1

AND gate and logic "1" drops for the OR gate. Only a limited number of series cascaded circuits can be used before the logic levels must be restored to the rails. Even more serious is the voltage degradation for the cascaded AND-OR function.

Figure 2. Cascaded diode logic showing level shifts. The obvious way to restore the logic level voltages is to use a double transistor inverter. However, it does not take a genius to notice the a single inversion on the end of each gate will make a NAND or a NOR gate and the logic level is automatically restored for each gate. The NAND gate is used extensively. As you no doubt recall, any logic expression can be implemented using just NAND gates. Thus, other forms are not necessary. DIODE TRANSISTOR LOGIC CIRCUITS A typical DTL NAND gate is shown in Figure 3. Observe the diode AND function on the front end and the transistor NOT at the output end. The extra resistors and diodes are used to maintain appropriate currents, to maintain proper functioning, and to guarantee certain noise margins. We will completely analyze this circuit, but will be particularly interested in the "terminal" characteristics of the gate. Our ultimate goal is to understand the operation of the gate so that the manufacturer's specifications can be understood.

Figure 3. DTL Circuit ANALYSIS OF THE DTL GATE Analysis of the DTL gate is dependent on complete understanding of the currents within the gate under all logic conditions. First let us develop a generic understanding of the operation. Of particular importance will be the direction of currents at the terminals of the gate. As in most logic systems, the transistor will either be cutoff or saturated.

Diode-Transistor Logic

2

If all inputs are high, (+5v), no current will come out of the input diodes at the input and current will flow down through the first 5K resistor and through the diodes D1and D2 toward the base of the transistor. Some current will split off and go down through the lower 5K resistor to ground. However, most of the current will go into the base of the transistor causing it to saturate, pulling the output low, VO =0.2 Volts. We will show this condition quantitatively shortly. If one or more of the inputs to the gate are held low (0.2 V), then the current down through the 5K resistor will go out the input diode, away from the transistor base. Under this condition, the transistor will be cutoff and the ouput will be high with VO =5 Volts. ANALYSIS WITH INPUT LOW Quantitatively, we will start with one or more inputs held low, at 0.2 Volts. From the logic function of the NAND gate, we know that the output is supposed to be high. Therefore, the transistor must be cutoff. To begin with, we will assume the two diodes D1 and D2 in series will also be cutoff. All the current coming down through the 5K resistor must all go out through the input diode, causing it to be on. The circuit with the models indicated is shown in Figure 4. From this circuit we can calculate all voltages and currents and prove (or disprove) our assumptions about the condition of each element.

Figure 4. DTL gate model with input low The voltage at point P is VP = 0.2 + 0.70 = 0.90 Volts. We need to show that this voltage is low enough that the two series diodes and the transistor will be cutoff. The argument is that if either diode carries current, then both must. Since 0.9 Volts is not enough across the pair to maintain conduction, then neither conducts. Given that the diodes are off, then the voltage at the base of the transistor is zero, and is also cutoff. We can verify that the input diode is conducting by observing that current I1 is I1 = (5-0.9)/5K = 0.82 mA

Diode-Transistor Logic

3

This current leaves through the input diode, hence it is on. The current entering the input terminal is I in = - I1 = -0.82 mA The negative sign occurs because the input current is defined as going into the terminal. We can now verify that the output voltage is 5 Volts because the transistor has been shown to be cutoff. Thus, VO = 5.0 Volts One other characteristic of the DTL gate that can be obtained at this point is the range of input voltages that will be recognized as a "low". From the logic function of the NAND gate, this can be translated into the question of how high the input voltage may rise and still keep the transistor cutoff. The transistor will remain cutoff as long as the voltage at the base does not rise above 0.5 volts. At this voltage, there will be current down through the lower 5K resistor to ground. This current must come from the +5 supply down through the upper 5K resistor and the diodes, D1 and D2. Hence the diodes must be conducting. This current will be 0.1 mA. The voltage at point P will be VP = 0.5 + 0.7 + 0.7 = 1.9 Volts The current I1 is I1 = (5-1.9)/5K = 0.62 mA The current going out through the input diode will be Iin = -(0.62 - 0.1) = -0.52 mA indicating that the input diode is still conducting. Figure 5 shows the resulting circuit with the circuit models included. The maximum voltage at the input that is guaranteed to be recognized as a low is VinLmax = 1.9 -0.7 =1.2 Volts. This result is included in the table of terminal specification at the end of this chapter.

Diode-Transistor Logic

4

Figure 5. DTL circuit model with Vin = VinLmax ANALYSIS WITH ALL INPUTS HIGH When all inputs are high, all current down through the upper 5K resistor will go toward the base of the transistor, causing it to saturate. The series diodes will obviously be conducting, and we will show that the input diodes are cutoff. Figure 6 shows the circuit with these models.

Figure 6. DTL circuit model with inputs high The voltage at point P is VP = 0.8 + 0.7 + 0.7 = 2.2 volts Thus, I1 is I1 = (5-2.2)/5K = 0.56 mA This current will go through the diodes toward the base of the transistor. Some of the current will go down through the lower 5K resistor with the rest going into the base of the transistor. With the transistor saturated, the current going down through the 5K resistor will be I2 = 0.8/5K = 0.16 mA The base current then is IB = 0.56 - 0.16 = 0.4 mA Diode-Transistor Logic

5

If the transistor is to be saturated, the maximum collector current is ICmax = βIB = 30 * 0.4 mA = 12.00 mA at saturation, the current coming down through the 2.2K collector resistor is I3 = (5-0.2)/2.2K = 2.182 mA this current is much less than the maximum saturation current and we see that with no load, the transistor will, indeed, be in saturation. In fact, there is excess capacity in collector saturation current. This excess capacity can be used to sink external load current. This current is called Io or load current. The maximum load current this gate can sink is IoLmax = 12.00 mA - 2.182 mA = 9.818 mA Note that this current is entering the terminal of the gate, hence, is positive. CALCULATION OF VinHmin and IinH We need to go back now and look at the input voltage in Figure 6 and determine the minimum allowable input voltage that will still be recognized as a high, VinHmin. There are several ways we could define this value, but we will use a straightforward definition. That is, we will define the input to be high as long as no current flows through the input diodes. In other words, as long as the input diodes are cutoff. Thus, VinHmin = VP - 0.60 = 2.20 -0.60 = 1.6 Volts. Since the input diodes must remain cutoff, IinH = 0. CALCULATION OF FANOUT If several load gates are connected to the output terminal of the gate we are looking at, we need to look at the current output drive capability compared to the input current requirements of the load gates. Because the input current is zero when high, an infinite number of load gates can be driven when high. However, the DTL gate requires current when the input is low. This situation is shown in Figure 7. We made that calculation earlier and found IinLmax = - 0.82 mA. Note that this current is negative meaning that it is coming out of the input terminal. We also found the output of a gate can sink 9.818 mA when it is low. We can now calculate the fanout.

The maximum number of gates that can be driven as loads is 11. Fanout = 11 Diode-Transistor Logic

6

Figure 7. DTL driver gate with N identical DTL load gates We also need to check voltage compatibility. That is, VoH>VinHmin and VoL IC2 , and, therefore, Q2 is saturated. The current coming out of the emitter of Q2 is the sum of the base and collector currents. Part of this current will go down through the 1 KΩ resistor to ground and the rest will enter the base of Q3. IB3 = IB2 + IC2 - I3 = 0.675 + 2.857 - 0.8 = 2.732 mA

Transistor-Transistor Logic

3

The maximum collector current that Q3 can carry and still be in saturation is βIB3 =81.96 mA, assuming β=30. The maximum current the gate can sink when the output is low IoLmax = ICsatmax - I4 = 81.96 - 1.2 = 80.76 mA Now let's turn our attention back to the input and determine VinHmin and IinH . We will define the input voltage to be high as long as no current goes out the input terminal. Thus, all we have to do is keep the input voltage high enough so that the B-E p-n junction of Q1 does not turn on. Thus, VinHmin = 2.3 - 0.6 = 1.7 Volts CALCULATION OF IinH With the input voltage at a high, say 5 volts, the transistor Q1 will be operating in the reverse active mode. The B-E junction is reverse biased, and the B-C junction is forward biased with a base current of 0.675 mA. If there were significant curent gain, you would expect to see a large current going into the input. However, the reverse β is typically on the order of 0.02. Thus, IinH = βR ∗ I = 0.02 ∗ 0.675 = 0.0135 mA This current would add to the current going into the base of Q2, but is ignored because it is quite small and because βR is made as small as possible and this input current is a maximum and cannot be counted on. THE TOTEM POLE OUTPUT STAGE One of the problems with the TTL gate circuit we have been analyzing is that the pull-up resistor on the output transistor will prevent rapid charging of any wiring capacitance on the output. One way to improve the rise time is to reduce the resistance value as is often done, but this also increases the power dissipation when the output is low. If we look at the circuit, we observe that when the transistor is saturated, it presents a very low effective resistance to ground. The problem arises when the output is high and the pull-up resistor is too large. Ideally we would like to have a very low resistance pull-up when the output is high, but a very high pull-up resistance when the output is low. In this way, we could get quick charging and very low power dissipation. The totem-pole output stage for TTL, shown in Figure 6, does just that.

Figure 6. TTL gate with totem-pole output.

Transistor-Transistor Logic

4

This circuit operates just like the original circuit except that Q4 is on when the output is high and off when the output is low. We need to verify this operation. OUTPUT LOW Figure 7 shows the TTL circuit with all inputs high and the output low. The models for the transistors are shown as before, except diode D and transistor Q4 are added and shown as cutoff.

Figure 7. TTL gate with totem-pole output circuit model with inputs high. The analysis of this circuit proceeds exactly the same as before. The currents, I1, I2, I3, and IB3 are the same as before. With the diode and Q4 not conducting, IoLmax is now the same as IC4max , 81.96 mA. We only need to show that the diode D and transistor Q4 are indeed off. The voltage at the bottom of the diode is 0.2 Volts and the voltage at the base of Q4 equal to the voltage at the collector of Q2; VC2 = (0.2 + 0.8) = 1.0 Volts. Thus, the voltage across the B-E junction of Q4 plus the diode is 0.8 Volts. If one conducts, the other must also. To take both out of cutoff would require at least 0.5 + 0.6 = 1.1 Volts. Thus, both are off.

OUTPUT HIGH This condition occurs when one or more inputs are low. The circuit is shown in Figure 8 with the appropriate models used for the transistors and the diode. In this case, Q2 and Q3 are both cutoff while Q4 and the diode are conducting. We have to assume here that there is some load and that the output current is not zero.

Transistor-Transistor Logic

5

Figure 8. TTL totem-pole circuit model with output high. The current coming out the output terminal IS (=-Io ) is the sum of the currents coming down through the base and the collector. Thus, IS = IB4 + βIB4 Because each TTL load represents 13 µA, if we assume there are 10 loads, then IS = 130 µA. The base current is I B4 =

130µA = 4.2 µA 1+ β

where we have assumed a β of 30. Then taking the path down through the 1.4 KΩ resistor to the output. the output voltage is Vo = 5.0 - 4.2µΑ∗1.4K - 0.7 -0.7 = 3.6 Volts The voltage drop across the 1.4 KΩ resistor is neglegible. Of course as the current increases, the output voltage will drop further. TERMINAL SPECIFICATIONS OF THE TTL GATE We are now ready to make the table showing the terminal specifications for the TTL gate. These are shown in Table 1. -----------------------------------------------------------------------------------------------Table 1. Terminal Specifications For TTL VinLmax = 1.2 V

VoL = 0.2

VinHmin = 1.7

VoH = 3.4 (@Io = -130 µA)

IinL = -1.025 mA

IoLmax = 81.96 mA

IinH = 13 µA

IoH = undetermined

Transistor-Transistor Logic

6

Figure 9. Data for ‘00, ‘04, ‘10, and ‘30 NAND gates for several TTL families (Abstracted from Texas Instruments TTL Data Book.)

Transistor-Transistor Logic

7

MANUFACTURER’S DATA SHEETS The terminal specifications of several TTL families are shown in Figure 9. You will note the values given for various voltages and currents are quite different from those we calculated. This difference comes from the fact that manufacturing tolerances and variations cannot be closely controlled, hence, the specifications given by the manufacturers are much more conservative than our calculations which were based on nominal values. Also note that the limits are usually given as a maximum or a minimum, depending on which limit is normally used in design. For example, IinLmax is given as -1.6 mA for the 74xx series. What this means is that as a designer, your driver must be able to sink as much as 1.6 mA when the input to the gate is pulled low. You will note that the TTL gate is rather loosely specified. The question invariably arises as to how one reads the data sheets or designs with this data. Figure 10 shows the allowed operating regions for a 7400, 2-input NAND gate. The best description of these operating regions is probably given by the following examples. Example 1: If the input voltage is between 0.00 and 0.08 volts, the output voltage will be below VCC and above 2.4 volts as shown in Figure 10a. Example 2: If the input voltage is between 2.00 and VCC, the input current will be between 0 and 40 µA as shown in Figure 10b. You will note in the above examples that there is no mathematical relationship between one variable and another. There is simply not enough data to develop one and the variability of the manufacturing process prohibits the manufacturer from providing one. In a design setting, you must stay within the limits provided by the manufacturer. For example, if you wanted to connect a resistor from the output of a 7400 gate and ground, what would be the limits allowed on the resistance value? Figure 10c provides part of the answer. If we assume that we must operate within the shaded region which represents a “High” level output, we should not allow the output to drop below 2.4 volts with 400 µA coming out of the gate. The minimum value would be 6 KΩ. The upper limit is, of course, infinite; an open circuit (Vo = VCC, Io = 0). There are a few data points provide by the specifications that are not within the limits of normal operation. For example, if the input voltage drops below zero, it is allowed to drop to -1.5 volts where you may expect as much as 12 mA coming out of a 7400 gate input. Most of this current comes from an input clamp diode which has not been shown on our drawings.

Transistor-Transistor Logic

8

Transistor-Transistor Logic

9

Another non-standard data point of interest is IoS, the short-circuit output current. This is the current you get from the gate output if the output is shorted to ground when the output of the gate would otherwise be high. In this case, both minimum and maximum values are given, -18 to -55 mA. If you went to the laboratory and actually performed this deed, you could expect a current somewhere in this range. How does this affect the designer? For example, a designer might be tempted to connect the output of the 7400 gate directly to the base of an NPN transistor whose emitter is grounded. In this case, the "high" output voltage is clamped at 0.80 V by the BE junction. What current can you expect into the base of the transistor when the gate output goes "high"? This condition is tricky and perhaps open to some debate, but the conservative designer must recognize that the operation is between the short circuit case and the case where VoHmin=2.4V when IoH =400µA. The conservative designer would conclude that the current might be as low as 400 µA and as high as 55 mA; the worst cases. It is possible to go back to the circuit of the gate with 0.8 volts at the output terminal and calculate the current. However, this analysis would be for nominal values only and not provide definitive limits on the current. Note the notation at the bottom of the specification table in Figure 9. This notation discusses the limit on the amount of time a short circuit is allowed to be connected to the output of some gates. This time limit is based on the amount of time it takes the internal components of the integrated circuit to heat up to its maximum allowed value. While connecting a transistor base to the output is not exactly a short circuit, it is outside the allowed operating region and probably should have the same time limits as the short circuit. TERMINAL CHARACTERISTICS During the previous discussions on TTL, we were looking at circuit operation and developing an understanding of how the terminal specifications were arrived at. Let us now take a broader look at these characteristics. First, the input currents are quite high when the input is low, requiring the driver to sink a lot of current. When the input is high, the input current into the gate is quite low. Thus, any circuit which is supposed to drive the input to a TTL gate must concentrate on sinking current, and only needs to source a little current when the driver output voltage is high. Second, the output strength of the TTL gate matches the strength requirements at the input. An example is given in Figure 11. The TTL gate can sink a large current when its output is low, but can only source a small current when the output is high. Thus, if the TTL gate is expected to drive a circuit that is not another TTL circuit, you must exercise care when designing the interface. The load circuit must not require large input currents when its input is high, but may use larger currents when the input voltage is low.

Transistor-Transistor Logic

10

Figure 11. When driving other TTL gates as loads, a 7400 gate must be able to sink more current than it needs to source. These requirements must be kept in mind when designing interfaces with the TTL gate at both the input and the output. Examples of interfacing with TTL gates are shown in Figure 12 and 13. See data books for more complete data.

Transistor-Transistor Logic

11

Figure 12. Several ways to drive loads from TTL gates.

Figure 13. Several interfaces to drive TTL gates.

Transistor-Transistor Logic

12

WIRED-AND CONNECTION Because the active pull-up or totem-pole output of the TTL gate always has one transistor cutoff and the other turned on, you cannot connect two outputs together. If one is trying to pull the output high, and the other is trying to pull it low, you will have a very low impedance path to ground and very large currents. For the same reason, the output must not be connected to any voltage source or to ground through a low impedance path. In one state or the other, there would be a low impedance path and large currents. OPEN COLLECTOR GATES In order to overcome the limitations created by the totem pole output circuit, some gates are manufactured with the output collector left open. One example is the 7405, a quad 2input NAND gate with open collector outputs. If you connect a resistor as the pull-up, you can use this resistor to source current when the output is high and/or you can wireAND the collectors together. TTL FAMILIES As the designers of TTL gates became more sophisticated, they developed modifications which would provide special characteristics. The original series of TTL was designated as 74XX, where the XX is replaced by logic function ( 00 is a quadruple 2-input NAND, 04 is a hex inverter, etc.) The 74LXX series is a low power family. 74HXX is a high speed family. 74SXX is a family based on Schottky diodes and transistors. 74LSXX is a family of low power Schottky. A 54xXX is also provided as a companion family to the 74xXX families. The 54... families are identical to the 74... families, except for operating temperature range and tolerance on power supply voltage. Each family has different characteristics, but the same logic functions. The L family is low power, but is much slower than the standard family. The H family is high speed, but also has higher power dissipation. The Schottky families are quite fast without increasing the power dissipation. More recent advances in TTL family have given us several other versions. For example, 74F, 74AS, and 74ALS, for Fast, Advanced Schottky, and Advanced Low-power Schottky. The AS family is the fastest, with a propagation delay of less than 5 ns. Table 2 shows the propagation delays and power supply current for each type of gate. The power supply current, ICC, is the average for a 50% duty cycle with the output spending half its time low and half the time at a high. In addition, these different families use slightly different circuit configurations. A little study of the circuits will reveal the same operations.

Transistor-Transistor Logic

13

Table 2. Propagation delays and power supply current for TTL families Data abstracted from Texas Instruments TTL Data Books

Gate

tPLH (ns) typ max 11 22 35 60 5.9 10 9 20 2 3 4.5 3 11 1 5

min 7400 74L00 74H00 74LS00 74S00 74ALS00 74AS00

tPHL (ns) typ max 7 15 31 60 6.2 10 10 20 2 3 5 2 8 1 4

min

ICC (mA) typical 2.00 0.20 4.50 0.40 3.75 1.00 6.20

SCHOTTKY TTL A Schottky PN junction is made up of a semiconductor and a metal. This kind of junction has two characteristics: low turn-on voltage and low junction capacitance. When a Schottky junction is used in place of or in parallel to the Base-Collector junction of a transistor, the transistor is faster because of the lower junction capacitance and because the transistor cannot go so deep into saturation. Because the turn-on voltage for the BC junction is lower, VCEsat is higher. Schottky TTL is thus faster than standard TTL and the terminal voltages are slightly different. See the data sheet. TRI-STATE OUTPUT The totem-pole output of a TTL gate provides additional speed at lower power for the gate than a simple pull-up resistor. The cost, however, is that the gate outputs cannot be connected in parallel. This problem is serious when you need to make a bus structure such as a data bus, where several gates need to put data onto the bus at different times. The outputs can be OR'd or AND'ed using appropriate gates, but this solution is less than satisfactory and slows down the operation. A better solution is the TRI-STATE output as shown in Figure 14. The added input allows normal operation of the gate when the "enable" input is high. Both output transistors are cutoff when "enable" is low.

Transistor-Transistor Logic

14

Figure 14. TTL gate with Tri-State output. OTHER LOGIC FUNCTIONS IN TTL TTL integrated circuits have been manufactured that perform over one hundred different functions. Many are small-scale integrated circuits that perform simple logic functions in addition to the basic NAND as well as flip-flops. Others are medium-scale integrated circuits that perform more sophisticated functions such as encoders, decoders, and registers. The more complex functions are implemented in TTL using combinations of the basic logic gates. It is of interest to look at how some of the other logic funcitons are implemented. Figures 15, 16, and 17 show a NOR gate, an AOI gate (AND-ORINVERT), and an AND gate. In each case, the input and output circuits are the same as for the NAND gate. Extra parts are added to perform the desired function. For example, the NOR gate has extra Q1 and Q2 added in parallel to the existing transistors. The extra emitters in Q1 are left out. These extra emitters perform the AND function in the NAND gate. The OR is performed by the Q2s in paarallel. The final inversion is performed by the output stage by Q3. Note the similarity to the AOI gate in which the AND function is intact with the extra emitters in the Q1s. The AND gate in Figure 16 has an extra internal inversion. The actual inversion is done with Q5. The other additional transistor, Q6 is used simply to guarantee Q2 is turned off when Q5 is turned on. This extra transistor is necessary to maintain voltage levels consistent with the NAND gate.

Figure 15. TTL NOR gate implementation

Transistor-Transistor Logic

15

Figure 16. TTL AND-OR-INVERT (AOI) gate implementation

Figure 17. TTL AND gate implementation How do these modifications in the basic TTL circuit affect propagation delays? Table 3 show the typical and maximum propagation delays for each type of gate discussed. It is apparent from the table that the addition of the extra internal inverter stage in the AND gate does have a significant effect on propagation delays while the addition of the parallel circuits for the NOR and AOI gates has very little effect. This latter effect is caused primarily by the addition of internal capacitance. Table 3. Propagation delays for NAND, NOR, AND, and AOI gates. Data abstracted from Texas Instruments TTL Data Book

Logic Gate 7400 (NAND) 7402 (NOR) 7408 (AND) 7451 (AOI)

tPLH (ns) typ 11 12 17.5 13

Transistor-Transistor Logic

tPHL (ns) max 22 22 27 22

typ 7 8 12 8

max 15 15 19 15

16

EXERCISES 1. For the TTL NAND gate circuit shown draw a wire connecting both inputs to +5 volts. Then: a. Indicate the output logic level: ________ b. Indicate the state of each transistor and the diode. Q1 _______, Q2 ________, Q3 ________, Q4 _______, D ______ c. On the diagram, draw an arrow showing the direction of current in each branch.

2. On the circuit below, draw wires connecting one input to ground and the other to +5 volts. Then: a. Indicate the output logic level: ________ b. Indicate the state of each transistor and the diode. Q1 _______, Q2 ________, Q3 ________, Q4 _______, D ______ c. On the diagram, draw an arrow showing the direction of current in each branch.

Transistor-Transistor Logic

17

3. For the TTL NOR gate circuit shown draw a wire connecting both inputs to ground. Then: a. Indicate the output logic level: ________ b. Indicate the state of each transistor and the diode. Q1a _____, Q1b_______, Q2a ____, Q2b________, Q3 ______, Q4 _____, D _____ c. On the diagram, draw an arrow showing the direction of current in each branch.

4. For the TTL NOR gate circuit shown draw a wire connecting the A input to +5 volts and the B input to ground. Then: a. Indicate the output logic level: ________ b. Indicate the state of each transistor and the diode. Q1a _____, Q1b_______, Q2a ____, Q2b________, Q3 ______, Q4 _____, D _____ c. On the diagram, draw an arrow showing the direction of current in each branch.

Transistor-Transistor Logic

18

5. Assume the following circuit is connected on a proto-board in the laboratory. You are troubleshooting the circuit. The gates are all 74LS00. a. What voltage do you expect at each node, A-E? b. What current would you expect in each branch, (beside each arrow)?

6. In most cases a resistor is used bewteen an input terminal and +5 volts to make a "1" at the input. Sometimes a resistor is used to pull-down the input to a zero. Both cases are shown below. What is the maximum value in each case? a. Rpull-up max _________

b. Rpull-down max ____________

7. Calculate the minimum resistor allowed in each case for the two 74LS00 gates below. a. Rpull-up min _________

Transistor-Transistor Logic

b. Rpull-down min ____________

19

Problems 1. You are in the midst of designing a combination portable computer/walkman/Internetcellphone which you intend to market and hopefully become a millionaire. To complete the design you need a 3-input NAND gate. The only store open at this time of night is the Handy-Combo Electronics/Oil Change/Hair Style/Pizza Parlor where you purchase the TTL gate shown below. Unfortunately, there are no specifications except that βsat =15 and βr =0.05. You must determine the specifications for this gate before you can use it. List the eight terminal current and voltage specifications in tabular form.

2. Make a compatibility chart for three TTL series shown in the data sheet in Figure 9 of the text. This chart is a matrix showing how many loads of a particular series can be driven by a driver of the same or another series, all of course, within specifications given by the manufacturer.

3. The two gates shown in the diagram labeled problem 3 are 74LS00. Gate A is used to drive gate B as well as the two-transistor buffer circuit. When the output of Gate A is high the load is turned off. When the output of gate A is low, the load is carrying current. Gate B should see valid logic levels and must operate properly. Determine the absolute maximum current that can be delivered through the load using the following characteristics for the buffer transistors. VBEcutin = 0.5 volts, VBEsat = 0.8 volts, VCEsat = 0.2 volts, Betasat = 20

Problem 3

Problem 4

4. In the above logic system for problem 4, the output connection for each of the eight gates is numbered. If you were to measure the current in each of these wires, what is the maximum value you would expect to find? Use the specifications in Figure 9. For the load resistors connected to outputs 6 and 7, give the allowable range assuming valid logic output voltage levels. The symbols inside the gate indicate the family. No symbol indicates a 7400 gate.

Transistor-Transistor Logic

20

5. Draw the figures showing the allowed operating regions, as shown in Figure 9 in the textbook for a 74S00 gate.

6. Laboratory Measurement For a 7400 gate, make the following measurments: a. No-load output voltage b. VinLmax c. VinHmin For the last two measurement, describe the criteria you used to determine the limits.

Transistor-Transistor Logic

21

7. A TTL gate has the following specifications: VinLmax =1.00 VinHmin =2.00 IinL = - 0.500 IinH = 1.00 VoLmax = 0.50 VoHmin = 2.50 IoLmax = 5.50 IoHmax = -7.00 In both cases below, the gate must see a Low with the switch in one position and a High with the switch in the other position. a. Determine the allowed values of resistor RD in the circuit below. b. Determine the allowed values of resistor RE in the circuit below.

Transistor-Transistor Logic

22

Chapter 7

EMITTER-COUPLED LOGIC The major speed limitation of TTL is the turn-off time of saturated transistors. To be sure, TTL has come a long way from the 100 ns time of DTL to the 2-4 ns propagation delays of ASTTL. ECL is designed so that the transistors are either cutoff or in the active region, rather than cutoff or saturation for the DTL/TTL. With the transistors in the active region, the charge stored in the base region of the transistors is kept to a minimum, allowing shorter turn-off times. Typical propagation delays of "standard" ECL are about 1 ns, and down to about 0.5 ns for some of the advanced types. There are several disadvantages associated with ECL. It uses a negative power supply so that the logic levels are not compatible with any other logic family, and makes analysis and measurement inconvenient. ECL requires large currents and the noise margins are small. On the other hand, power supply currents remain much more stable when the logic switches compared to TTL, thus reducing noise on the power leads. In practice, ECL is used only when necessary for its high speed. ECL has been around since the early 60's, being developed at about the same time as DTL. Before we start the analysis of the circuit, we need to look at the circuit model we will use for the transistors in the ECL circuit. Because ECL is predicated on speed, the size of the transistors is as small as possible to keep stored charge to a minimum. Thus, current densities and hence, voltage drops will be slightly higher than normal. For this reason, we will use different voltages for base-emitter voltage for ECL. VBEγ = 0.65 and VBEactive = 0.75 volts ECL is based on the emitter coupled pair shown in Figure 1. This pair will have one transistor on and the other cutoff in each logic state. If we compare the currents through the two transistors, one with 0.75 volts between base and emitter and the other with 0.65 volts, we will find that the transistor with the higher voltage will be carrying nearly 50 times the current of the other. Thus, the lower voltage transistor will essentially be cutoff. 0.75 VT

I B1 I o e = = 47.5 0.65 I B2 I o e VT

Emitter-Coupled Logic

where VT = 25.9 mV

1

Figure 1. Emitter coupled Pair BASIC OPERATION OF THE EMITTER-COUPLED PAIR We start by looking at the operation of the difference amplifier shown in Figure 1. The two transistors are connected at their emitters. The base of Q2 is connected to a reference voltage, VR. Consider for an example, that the input voltage is low enough to keep Q1 cutoff. Then, Q2 will be conducting and the emitter voltage will be VE = VR - VBEon = VR - 0.75 assuming Q2 is operating in the active region. We know Q2 is in the active region because the collector resistors are selected to keep the transistors from saturating. How high can we raise the input voltage and still keep Q1 off? VinLmax = VE + VBEγ = VR - 0.75 + 0.65 = VR -0.10 volts Now, let the input voltage start rising above this voltage. Then Q1 begins to turn on. This increases the current through RE, raising the voltage at the emitters. This voltage rise causes Q2 to begin to turn off. The input voltage rises only a little bit before Q2 turns off entirely. The minimum high level input voltage occurs when VE = VR -0.65 and VinHmin = VE + 0.75 = VR + 0.10 volts Thus, a 200 mV swing at the input causes a complete reversal in which transistor is turned on and which is turned off. ANALYSIS OF THE ECL GATE We now turn our attention to Figure 2, the complete ECL gate circuit. In this circuit we have added two emitter follower output stages. As we shall see later, these output stages serve the purposes of buffering the gate output, providing added fanout, as well as shifting the voltage levels.

Emitter-Coupled Logic

2

Figure 2. Complete ECL Gate OR OUTPUT --- Output high, Q2 off With Q2 off, the output circuit looks like that shown in Figure 3.

Figure 3. Equivalent Circuit With Q2 OFF. If we assume a very high β, we can ignore base current (I2 ≅ 0) but the transistor Q6 is in the active region so VoH = Vo2 ≅ -0.75 volts (Note that since the collector is at 0V, and the emitter is at -0.75V, the transistor is in the active region. This type of circuit is called an emitter follower. The output voltage is always VBEactive below the base voltage and follows the base voltage, even when the output goes low as we shall see next.)

Figure 4. Equivalent Circuit With Q2 ON.

Emitter-Coupled Logic

3

OR OUTPUT LOW - Q2 ON, Q1 OFF, The circuit for this case is shown in Figure 4. The emitter voltage is: VE = VR - 0.75 = -2.07 volts Thus, the current down through the emitter resistor is IE =

. −2.07 − ( −5.2) 313 = = 4.018mA 779 779

If we assume β is very large so base currents can be neglected, the collector current in Q2 will be the same as the emitter current. Thus, the voltage at the collector of Q2 (and the base of Q6 ) will be VC2 = VB6 = -IC2∗245 = -0.98

(Note that VC2 - VE = 1.09v. Q2 is in active region.)

Thus, the voltage at the emitter of Q6 will be VoL = Vo2 = -0.98 - 0.75 = -1.73 volts We can now plot the output at Vo2 versus the input voltage. The result is shown in Figure 5. NOR OUTPUT, Q1 OFF, Output High This case is essentially the same as for the OR output when it is high, except that we have a 220 Ω resistor instead of a 245 Ω resistor at the collector. Again, because the base current is so small, the output voltage will be -0.75 volts, the same as for the OR output. NOR OUTPUT, Q1 ON, Q2 OFF, Output Low. This case is similar to the OR output high case, except that the input voltage can directly affect the collector current through Q1 causing the voltage at the base of Q1 to change. As Vin increases, the current increases causing the voltage at the base of Q5 to decrease. Because the output follows the base voltage with a 0.75 volt difference, the output drops as Vin increases. This scenario holds until Q1 saturates. Then as Vin increases further, the base-collector junction is forward biased, and current flows from the input toward the collector, raising the voltage at the base of Q5, hence raising the output voltage. The resulting plot of Vo1 vs Vin is also shown in Figure 5.

Emitter-Coupled Logic

4

Figure 5. Voltage Transfer Characteristic For The ECL Gate

EXERCISES 1. For the 2-input ECL gate below, write a logic LOW at both inputs. a. Beside each transistor, write the state of the transistor. b. At the two outputs, write the logic level. c. Draw arrows showing the paths of current in the circuit.

2. For the 2-input ECL gate below, write a logic LOW at input a, and logic HIGH at input b. a. Beside each transistor, write the state of the transistor. b. At the two outputs, write the logic level. c. Draw arrows showing the paths of current in the circuit.

Emitter-Coupled Logic

5

For the following exercise, assume VBEγ = 0.65 volts, VBEact = 0.75 volts, and β is very large (assume base currents are negligible). 3. For the ECL gate below: Assume Both A and B inputs are low find: b. VinLmax = ______ a. VE = ______

c. IE = _________

d. I330 = _____

e. I340 = _______

f. VC2 = _______

g. VC1 = _____

h. VO1 = _____

i. VO2 = ______

4. For the ECL gate, assume input A is high and input B is low: b. VinHmin = ______ c. IE = _________ a. VE = ______ d. I330 = _____

e. I340 = _______

f. VC2 = _______

g. VC1 = _____

h. VO1 = _____

i. VO2 = ______

Emitter-Coupled Logic

6

Problems

1. An ECL gate is shown below. Sketch the Vo vs Vin characteristic for the OR output. Show the values at all breakpoints. Assume VBEact = 0.75 and the transistor is effectively non-conducting at VBE=0.65V. Also assume beta is very high so that base current can be neglected.

Emitter-Coupled Logic

7

Chapter 10 Advanced CMOS Circuits Transmission Gates NMOS Transmission Gate The active pull-up inverter circuit leads one to thinking about alternate uses of NMOS devices. Consider the circuit shown in Figure 1. Here we have an NMOS transistor with its substrate connected to ground rather than the source terminal. In effect, the transistor is symmetric and can conduct in either direction. In order to turn it on, the gate voltage must be higher than either terminal by the threshold voltage. To turn the transistor off, the gate voltage must be lower than both terminal voltages plus the threshold voltage. On: VG > Vin + VT or VG > Vout + VT

(VG - VT > Vin

or

VG - VT > Vout)

Off: VG < Vin + VT and VG < Vout + VT

(VG - VT < Vin

or

VG - VT < Vout)

Figure 1. NMOS Transmission Gate Consider the following three cases Case I: VG = 0, Vin = 0 or 5 volts, Vout = 0 or 5 volts In this case, the transistor will be cutoff because the gate voltage is not higher than either terminal. Whatever voltage is on the capacitor will stay there. In other words, the capacitor will hold its charge. Case II: VG = 5 volts, Vin = 5 volts, Vout = 0 volts (initially) In this case, the gate voltage is higher than Vout by more than the threshold. Thus, the transistor will conduct and the capacitor will be charged up. The output voltage will rise until it reaches Vo = VG - VT. Note that this is the same as the output voltage of the active pull-up NMOS inverter. Case III: VG = 5 volts, Vin = 0 volts, Vout = 5 - VT volts (initially) Here the gate voltage is higher than Vin so that the transistor is turned on and the capacitor will discharge. It will discharge all the way down to zero because VG > Vin + VT.

Advanced CMOS

1

Generalizing these cases, we can see that with the gate voltage low, the output voltage will not change. In other words, the gate is off. If the gate voltage is high, the transmission gate will be on and the output will follow the input voltage. In this case, the output voltage is constrained to be less than the gate voltage by the threshold voltage. It should be noted that all terminal voltages are assumed to be more positive than the substrate voltage. We normally assume that the gate voltage will switch between the power supply rails. PMOS Transmission gate the NMOS transmission gate has the disadvantage that the output voltage is constrained to be less than the gate voltage. We will look to see how the PMOS gate in Figure 2 compares.

Figure 2. PMOS Transmission Gate In this case, the substrate is connected to VDD so that the gate will turn on if the gate voltage is low and off when the gate voltage is high. The conditions for the PMOS transmission gate are: On: VG < Vin - VT or VG < Vout - VT Off: VG > Vin - VT and VG > Vout - VT We look at the same three cases Case I: VG = 5, Vin = 5 or 0, Vout = 0 or 5 volts In this case, the transistor will be cutoff because the gate voltage is not lower than either terminal Case II: VG = 0 volts, Vin = 5 volts, Vout = 0 volts (initially) In this case, the gate voltage is lower than Vin by more than the threshold. Thus, the transistor will conduct and the capacitor will be charged all the way up to 5 volts. The transistor always stays on because the gate voltage always remains lower than the input voltage by more than the threshold voltage.

Advanced CMOS

2

Case III: VG = 0 volts, Vin = 0 volts, Vout = 5 volts (initially) Here the gate voltage is lower than Vout so that the transistor is turned on and the capacitor will discharge. It will only discharge down to VT because at that point, Vout = VG + VT, which turns the transistor off. The PMOS has a similar, but symmetric characteristic to the NMOS when used as a transmission gate. Next we combine the two to develop an ideal transmission gate. CMOS Transmission Gate A CMOS transmission gate is shown in Figure 3. Included in the gate is an inverter (CMOS, of course) so that the NMOS and PMOS gates get complementary signals. When the control voltage, VC is low, the NMOS gate gets a low voltage and is turned off. The PMOS gate gets a high voltge and is likewise turned off. When the control voltage is high, the gate is turned on with both transistors able to conduct. As discussed above, each cannot conduct all the way to both power supply rails, but they do so at either end so they complement each other and together they can conduct to the rails.

Figure 3. CMOS Transmission Gate

Transmission Gate Resistance When the gate is off, the resistance of the gate is theoretically infinite. However, there is always some conduction so in reality, the resistance is finite. The off resistance is given in the specifications for the device and is typically in the megohms. When the gate is on, however, we would like the resistance to be low. To examine this case, consider the circuit in Figure 4. Here we look at the case where the input voltage is VDD, and the gate is turned on with the output voltage starting at zero. We look at the resistance as the capacitor is charged and the output voltage rises to VDD. The voltage across both devices is VDD - Vo, and the resistance of each is simply the voltage across it divided by the current through it, IDS for the NMOS of ISD for the PMOS.

Advanced CMOS

3

R=

VDD − Vo I

The total equivalent resistance is the parallel combination of the the two.

Figure 4. CMOS transmission gate used to calculate resistance. We start with Vout = 0. Here, the NMOS is in pinchoff IDS = kn(VDD - VT)2 Rn =

VDD − Vo k n (VDD − Vo − VT ) 2

The PMOS is also in pinchoff ISD = kp(VDD - VTp - 0)2 = kp(VDD - VTp)2

Rp =

VDD − Vo k p (V DD − VTp ) 2

The NMOS stays in pinchoff up until Vo = VDD - VTn at which time it cuts off. The PMOS, however, switches to triode when the output voltage rises to VTp. At this point, VSD is VDD - VTp and VSG = VDD - 0 = VDD. The criterion for triode region is VSG - VTp < VSD Thus, when Vo rises to VTp, the PMOS switches to triode and ISD = kp(2(VDD - VTp)(VDD - Vo) - (VDD - Vo)2)

Advanced CMOS

4

V Rp =

RP =

DD

−V o

k ( 2 (V −V )(V −V ) − (V −V ) 2 ) p DD Tp DD o DD o 1 k ( 2 (V −V ) − (V −V )) P DD Tp DD o

We have defined three distinct regions. If we assume VDD > VTp + VTn, the regions are as shown in Figure 5.

Figure 5. The three regions of operation for the CMOS transmission gate in positive transition Combining the two resistances in parallel for each region, we can plot the equivalent resistance over the transition at the output from 0 to VDD. The results in Figure 6 are ploted for the device parameters shown. The three curves are for the NMOS device, the PMOS device, and the parallel combination. The combined resistance is nearly constant over the entire region, a characteristic very helpful when designing. In this case, the designer usually assumes the device can be modelled by a fixed resistance when on and a very high resistance when off.

Figure 6. Transmission gate resistance as a function of output voltage.

Advanced CMOS

5

Logic Using Transmission Gates For the purposes of simplicity, we will represent a CMOS transmission gate with the symbol in Figure 7. This device uses 2 transistors

Figure 7. Symbol for Transmission Gate Many times, logic can be performed using transmission gates more efficiently than using standard CMOS logic gates. Efficiency in this case is the number of transistors. For example, the two input multiplexer using 2-input CMOS gates uses 10 transistors while it uses only 6 transistors using CMOS transmission gates as shown in Figure 8.

Figure 8. Two implementations of a 2-input multiplexer In Figure 8, We assume the CMOS 2-input NAND gates each take 4 transistors and the CMOS inverter takes two transistors. In the transmission gate implementation, we assume the transmission gate takes two transistors for the gate and two transistors for the inverter. We are assuming that the inverters in the transmission gate implementation can be shared on a single integrated circuit. The implementation of an exclusive or function is shown in Figure 9. The CMOS logic gate implementation takes 16 transistors while the transmission gate implementation takes 8 transistors.

Advanced CMOS

6

Figure 9. Logic gate implementation and transmission gate implementation of an XOR. Not all functions can be made more efficient using transmission gates. For example, a logic AND and a logic NAND made with transmission gates are shown in Figure 10. A CMOS AND requires 6 transistors while a CMOS NAND requires only 4 transistors. The transmission gate implementations require 4 and 6 respectively.

Figure 10. Transmission gate implementation of logic AND and NAND. BiCMOS

The major advantages of CMOS logic is its low power dissipation, simple fabrication process, and their abiltity to be made extremely small in size. Because CMOS gates are made as small as possible, their current drive capability is quite low and thus their greatest limitation is their limited ability to drive capacitive loads. This limitation becomes serious when implementing large scale memories or when driving loads external to the integrated circuit chip. The BiCMOS circuit was devised to take advantage of the bipolar transistor’s high current drive while still minimizing static power dissipation. A typical BiCMOS driver circuit is shown in Figure 11 with a capacitive load.

Advanced CMOS

7

Figure 11. Simple BiCMOS inverter. In the simple inverter in Figure 11, Q1 and Q2 are NPN bipolar transistors. Q3 is a pchannel MOSFET, while Q4 is an n-channel MOSFET. When the input voltage is low, the n-channel is turned off and Q2 is turned off. The p-channel is turned on. The base of Q1 is pulled up toward VDD and thus, turns Q1 on which in turn charges the load capacitor and drives any other high-level loads. When the output voltage rises high enough (VDD VBEγ), Q1 will be turned off. When the input is high, Q3 and Q1 are both turned off, while Q4 is turned on. A high voltage at the load will cause current through Q4 which causes the voltage at the base of Q2 to rise up and turn on Q2. Thus, the load capacitance can be quickly discharged through Q2. When the load voltage drops low enough, Q2 will turn off and the load continues to discharge through Q4. In both cases, when the load capacitance has been charged or discharged sufficiently, the bipolar transistors are turned off and there is no steady state current, preserving the low static power dissipation of CMOS logic. When the input is high, Q3 and Q1 are turned off. The resistance to ground allows any excess base charge to leak off to ground, allowing the transistor Q1 to turn off quickly. Similarly, the lower resistor can turn off Q2 quickly when the input is low. These turn-off times are important when there is a static load, and are not normally a problem with fully charged or discharged capacitive loads. The circuit shown in Figure 12 uses n-channel MOSFETs instead of resistors to provide even quicker discharge of excess base charge and thus, faster switching.

Advanced CMOS

8

Figure 12. BiCMOS inverter circuit. Reference: CMOS Digital Integrated Circuits, second Edition by Kang and Leblebici, McGraw-Hill, 1999 Exercises

1.

A simple NMOS transmission gate is shown for Problem 1. VTn = 1 volt, kn = 0.4 mA/V2, VDD = 5 volts To make the transmission gate conduct (turn it on) the control voltage should be: (High, Low) With the gate on, what is the output voltage when the input voltage is VDD? Vout = ________

Exercise Problem 1.

Advanced CMOS

Exercise Problem 2.

9

2.

A CMOS transmission gate is shown for Problem 2. VTp = VTn = 1 volt, kp = kn = 0.4 mA/V2, VDD = 5 volts To make the transmission gate conduct (turn it on) the gate voltage for the nchannel should be: (High, Low) To make the transmission gate conduct (turn it on) the gate voltage for the pchannel should be: (High, Low) With the gate on, what is the output voltage when the input voltage is VDD? Vout = ________ What is the current through the p-channel? Ip = ________ What is the current through the n-channel? In = ________

3.

The BiCMOS circuit shown as problem 3 below has a resisive load and the input is at 0 volts. What is the state of each device in the circuit? Q1 _____, Q2 _____, Q3 _____, Q4 _____, Q5 _____, Q6 _____ Draw all currents in the circuit. Write an equation for each current in the circuit in terms of the device parameters: β, VBEact, VBEsat, kn, kp, VTn, VTp, VDD All equations should be written in terms of these parameters and Vo. Write the nodal equation necessary to solve this circuit. If the load was capacitive instead of resistive, write the nodal equation.

Advanced CMOS

10

Problem 3 4.

Problem 4

The BiCMOS circuit shown as problem 4 above has a resisive load and the input is at VDD. What is the state of each device in the circuit? Q1 _____, Q2 _____, Q3 _____, Q4 _____, Q5 _____, Q6 _____ Draw all currents in the circuit. Write an equation for each current in the circuit in terms of the device parameters: β, VBEact, VBEsat, kn, kp, VTn, VTp, VDD All equations should be written in terms of these parameters and Vo. Write the nodal equation necessary to solve this circuit.

5.

For the BiCMOS gate shown for Problem 4, the resistor is replaced by a capacitor to ground. The initial voltage on the capacitor is VC(0). Rewrite the nodal equation to needed to solve the circuit.

Advanced CMOS

11

Chapter 11 Analog-Digital Conversion One of the common functions that are performed on signals is to convert the voltage into a digital representation. The converse function, digital-analog is also common. We will start with the digital-analog, D/A conversion. A simple D/A conversion circuit is shown in Figure 1. This circuit is essentially an operational amplifier inverting, summing circuit followed by a unity gain inverter. Each switch connects to zero volts, or +VR. The output voltage is Vout = Rf(V0/R0 + V1/R1 + V2/R2 + V3/R3 ...+ VN-1/RN-1) where V0 is the voltage at the switch 0, which is either +VR or 0, etc..

Figure 1. Simple digital to analog converter This representation is compared to a binary representation of a number with N bits. Binary Value = 2N-1aN-1 + ... + 23a3 + 22a2 + 21a1 + 20a0 where a0 can take on either 0 or 1, etc. If N = 8, then the binary number can take on any value between 0 and 255. In the D/A circuit, there would be 8 switches and input resistors. We want 256 discrete steps, 0-255. The smallest or least significant step is controlled by switch 0. The output voltage from

A-D Conversion

1

switch 0 is VR(Rf/R0), while the maximum output voltage is 255 times as high, when all switches are connected to VR. Thus, R0 = 2R1, R1 = 2R2, etc., so that each switch contributes exactly twice as much to the output voltage as the previous switch, and R0 = 128R7. The total range is 0 - 255VR(Rf/R0). We know that operational amplifier outputs cannot exceed the rails, or power supply voltages, so 255VR(Rf/R0) must be less than the power supply voltage. This limitation determines the relationship between the resistors and VR. For example, if we wish to limit the range to 10 volts with VR = 5 volts, then Rf/R0 = 2/255, or R0 = 127.5Rf. To make numbers convenient in the binary system, we usually make R0 = 128Rf. The resistor ratios in the input string are related by factors of two so that most significant resistor, R7 = 1Rf. Resistor values used in operational amplifier circuits are typically in the range of 1-100 KOhms. Lower values require large currents while large resistor values may cause errors due to bias currents and offset voltages. With high quality operational amplifiers, it is possible to use this range of 1-128 without serious problems, but extending the number of digits may cause significant errors which may exceed the least significant values. One way to overcome the large range in resistor values is to use a ladder network or the R/2R network shown in Figure 2. In the ladder, only two values are used, R and 2R. The overall gain is controlled by Ra and Rb.

Figure 2. Ladder type D/A converter

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We will assume the op-amp has infinite input impedance and all switches are connected to ground so the resistance seen looking any direction from any node is 2R as seen in Figure 3. Here we see that the two resistors looking left and right from node V0, are 2R. The parallel combination is R, so that the resistance looking up from node V1 is also 2R. This process can be carried out down the entire string. Similarly, the same process is used to see that the resistance looking down from each node is also 2R.

Figure 3. Ladder network showing resistances seen looking away from each node If switch SN-1 is connected to +VR, the voltage at node VN-1 is 1/3VR. This voltage is amplified by the op-amp as seen in Figure 2. Similarly, the voltage V3 = 1/3VR when switch V3 is up. However, the voltage VN-1 = 1/2V3. Thus, the switch S3 contributes only half the voltage to the op-amp. Similarly, each switch above, contributes successively less voltage by factors of two at each step. Up to now, we have simply shown switches in the A/D converters. However, solid state switches are universally used. These switches are most commonly CMOS switches as shown in Figure 4. The control voltage ussed is the same as the switched voltages to guarantee low resistance connections with the MOSFETS in the triode region when on.

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Figure 4. CMOS switch used in A/D converters Exercises 1. For a 4-bit D/A converter as shown in Figure 1, assume VR is 5 volts, and we want the range to be approximately 0-5 volts. the op-amp feedback resistor is Rf = 2.2 kOhm. Determine the necessary values of the input resistors. What is the resolution, or the smallest step? What is the highest output voltage? 2. A four-bit ladder type D/A converter is to be designed. Assume VR is 5 volts, and we want the range to be approximately 0-5 volts. What is the necessary gain of the op-amp? Analog - Digital Conversion Analog - digital conversion is the opposite to the process described above. In this case we want to convert an analog voltage to a digital representation. We see this function performed in digital voltmeters where a voltage is converted to digital and then shown in LED or LCD numerical displays. If we assume the input voltage can take on any value in the allowed range, such as 0-5 volts, the A/D converter will interpret that range in discrete steps. For example, if we are using an 8-bit converter, there are 256 steps within the 0-5 volt range. Each step represents a voltage range of 5/256 = 0.0195 volts, or about 20 mV. This discreteness means that any voltage between 0 and 19.5 mV will be represented as 00000000, while between 19.5 and 39 mV will be represented as 00000001, etc. The most straightforward method of doing an A/D conversion is to use a series of voltage comparators as shown in Figure 5. Here a string of resistors generates a string of voltage references for the comparators. The output of each comparator will switch to a high when the input analog voltage is above the reference voltage. There is a comparator for each step in the digital output above the zero step. Thus, in the case of an 8-bit conversion, there would be 255 comparators. These 255 signals would normally be encoded into the 8-bit digital representation. This method is very fast and is called a flash converter, but is impractical for all but the most critical applications where more than just a few steps are needed.

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Figure 5. Flash A/D converter A more practical, and more common, A/D converter using a counter and a D/A converter is shown in Figure 6. In this circuit, the counter starts at zero. The output of the counter goes to the D/A converter which presents its output to the voltage comparator. If the input voltage is above the output of the D/A, the counter counts up. As the conter counts up, the output of the D/A goes higher. When the D/A output goes above the input signal, the counter stops. At that point, the output of the counter is the digital representation of the input signal. The output of the voltage comparator can be used to tell the outside world when the conversion is completed. To take the next sample, the counter must be reset and the process starts over. The resolution is determined by the counter-D/A resolution and the range is determined by the range of output from the D/A converter. Of course, there must be a few more controls added such as stopping the counter when it reaches its maximum and an overflow signal enabled.

Figure 6. Counting A/D converter

Figure 7. Output of the D/A module

The output signal of the D/A converter module would look like Figure 7. The counter starts at zero which gives zero output from the D/A. The counter counts until the D/A output rises above the input voltage. The counting A/D converter does not resolve the conversion until the counter has counted far enough. Conceivably, this time could be 2N-1 counts. The maximum clock speed is

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limited to the time delays of the various modules; counter, D/A converter, comparator, and the AND gate. An 8-bit A/D with a 1 MHz clock could take 255 µs to resolve the output. This delay is very long compared to the flash A/D converter which would only take a few microseconds. We will look at two methods to try to speed this up; tracking A/D, and a successive approximation A/D Tracking A/D A tracking A/D converter uses an up/down counter and runs continuously. This type A/D converter is shown in Figure 8. The direction of the count is determined by the voltage comparator. Presumably, a change in the input analog voltage will be close to the previous voltage so the count does not take as long as a count from zero every time. This system has the added advantage of “tracking” the input voltage continuously. It also has the disadvantage that it continuously toggles around the input voltage.

Figure 8. Tracking A/D converter Successive Approximation A/D The successive approximation converter uses the same diagram as the tracking A/D converter shown in Figure 8. However, the binary counter use additional logic to control the count sequence differently. The counter switches the most significant bit to a 1 first. If the output of the D/A is lower than Vin, the bit stays a 1, otherwise, it is switched back to a 0 on the next clock pulse. Then the counter goes through the same sequence for the second most significant bit, etc., until all bits have been switched and compared. This type of A/D takes a maximum of 2N clock pulses to resolve the output, much faster than the maximum time of the simple counting A/D. Exercises 3. A simple counting 16-bit A/D converter uses a clock frequency of 2 MHz and has a range of 0-2 volts. What is the binary output when the input voltage is 1.5 volts? What is the settling time? 4. If the A/D converter in the previous problem is a successive approximation converter, what is the settling time?

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BCD Converters Typical digital voltmeters display the voltage in decimal representation. It would be appropriate for these voltmeters to maintain the digital representation in BCD to minimize encoding and decoding logic. The most logical way to handle BCD is to used BCD counters and BCD D/A converters in the feedback loop. BCD counters are well known, but we need to look at the coding of the D/A converters. In particular, we are interested in the resistor ladder networks that produce the BCD weighting. A simple BCD D/A converter is shown in Figure 9. In this case, the least significant decimal digit has resistance values a factor of ten time the corresponding resistor values for the most significant digit. An alternative implementation is given in Figure 10. In this case, the ratio between the digits is accomplished by the second summing inverting amplifier.

Figure 9. BCD D/A converter

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Figure 10. Alternative BCD D/A converter

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Chapter 12 Common Switching Functional Blocks Voltage Comparators In many applications, it is necessary to cause a digital switching action when an analog voltage rises above or drops below some value. An example would be a case when we wanted a digital signal to turn on a “discharge” light when the battery voltage dropped below a specific point, say 12.5 volts for an automotive application. In this case, we would want a logic high (or low) when the battery voltage dropped below 12.5 volts. We have seen how diodes and BJT base-emitter junctions as well as enhancement MOSFETs have thresholds where they beging to conduct. It would be possible to construct a circuit to create an output transition from low to high as the input voltage crossed the trigger point. As simple example would be an appropriate voltage divider to drop the battery voltage down to an appropriate level and feed it into the base of a transistor inverter followed by several more inverters to provide gain and make the output rail-rail transistion occur for a very small transition of the input signal. A few years ago, discrete devices were used as discussed to create this voltage comparison function. However, integrated circuits allow the voltage comparator function to be done in a single integrated circuit chip. These chips are based on operational amplifiers circuits with a switching transistor at the output. A functional representation is shown in Figure 1.

Figure 1. Functional representation of a voltage comparator and its symbol. This circuit is simply an operational amplifier followed by a transistor whose emitter is connected to ground and whose collector is open. A typical application is shown in Figure 2. In this case a reference voltage is connected to the Vin- input or the inverting input. When the signal voltage at the Vin+ input rises above the reference, the output voltage goes high. What’s really happening is that the output of the op-amp goes low, cutting off the transistor so the external pull-up resistor causes the output voltage to go high. When the input voltage goes below the reference, the output of the op-amp goes high, causing the transistor to saturate and pull the output low. If you reversed the connection of the two input signals, the output function would switch directions. All of the internal buffering and interfacing is taken care of within the op-amp part of the circuit.

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Figure 2. Typical application of a voltage comparator and its voltage transfer characteristic The VTC shows an abrupt switch when the input voltage passes the reference voltage. The transition is not instantaneous, but occurs over a few millivolts of the input signal for most comparators. The LM311 whose data is given on page 100 of the “‘Design Compendium”, shows a minimum gain of 40 V/mV, or a gain of 40,000. Thus, for the example above, if +V is 5 volts, the transistion occurs with a change of 0.125 mV at the input. The circuits shown above are typical of the LM319, and 339 voltage comparators. However, the LM311 is a little different in that the emitter of the output transistor is not connected to ground. It is left uncommitted, similar to the collector, so the user can have switching between other voltage ranges. An interface between positive voltage systems TTL or CMOS to the negative voltage ECL would be a typical application. It should be noted, however, that the emitter must be connected to a voltage within the +/- Vcc rails of the voltage comparator. Appropriate rails would be +5 and -5.2 volts of the two logic systems. One problem using voltage comparators with very slowly varying input signals is output signal oscillation when the inputs are in close proximity of the switch point. All signals including power supplies have a smal amount of noise embedded on top of the desired signal. We saw earlier that the switching transition occurs with less than a millivolt change in the input voltage. If, for example, the input signal was just at the switch point and there was a small noise blip on it, the output would switch. This switching can, and often does, induce more noise on the system. This additional noise can cause the input signal to drop, causing another transition in the output. This second transistion can then again induce noise reversing the transisiton again. Such oscillations are frequently seen and cause many problem in the circuits. Hysteresis One way to prevent the spontaneouus oscillation of voltage comparator circuits is to introduce hysteresis as shown in Figure 3. In this case, two additional resistors are added to provide feedback. Typically, R2 >> R3 so only a small amount of feedback is provided. As far as the comparator is concerned, the voltage at the non-inverting input is the voltage that it sees and will control the output state.

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Figure 3. Voltage comparator with hysteresis If Vin+ is just slightly below Vref, Vo ≈0, the output transistor is saturated. R2 Vnoninv = Vin + R2 + R3 As long as R2 >> R3, Vnoninv = Vin+ If we now raise the input voltage slightly so the output switches state, then R2 + R1 R3 Vnoninv = Vin + + V DD R3 + R2 + R1 R3 + R2 + R1 Again, as long as R1 + R2 >> R3 then the first term is approximately Vin+. However, the second term adds to the voltage at the non-inverting terminal. Because R3 is much smaller than the the sum of R1 and R2, this amount is quite small, but not zero. Typically, you want this amount to be a few millivolts to prevent oscillation. Example: Provide 10 mV of hysteresis with VDD = 5 volts, R1 = 1 kOhm, VREF = 2.00 volts. Solution: We want R2 >> R1 so we choose R2 = 1 MOhm. Then to get 10 mV hysteresis, R3 10mV = 5 R3 + R2 + R1 R3 0.01 = =.002 5 R3 + 1001 MOhm . R3 ≈ 0.002 x1 MOhm = 2 KOhm If we use R3 = 2 KOhm, the actual hysteresis is 9.97 mV.

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The voltage transfer characterisitc with hysteresis looks like that shown in Figure 4. Here we see that the high output transistion occurs at 2.00 volts as would have been the case without hysteresis, but the input voltage must drop significantly lower by the amount of hysteresis than before to cause the comparator to switch back.

Figure 4. VTC with hysteresis for example. Exercise Draw the voltage transfer characterisitc for the voltage comparator circuit in Figure 5. Assume the transistor acts as a perfect switch, open or short. Note in this case, the reference is on the non-inverting input to the comparator and the input voltage is on the inverting input. The reference is supplied by a voltage divider from the power supply. (Hint: Convert the reference supply to it Thevenin equivalent.)

Figure 5. Voltage comparator circuit for Exercise.

The 555 Timer The 555 timer is a functional block on a single integrated circuit chip that provides several timeing functions. The most common uses are as a square wave oscillator and as a source for a timed pulse or single-shot oscillator, astable and monostable multivibrators respectively. Timing and function are controlled by external componets and connections. Figure 6 shows the functional block diagram of the 555’s internal components. Comp A and compB are voltage comparators as discussed above. The three-5 KOhm resistor string provides two reference voltages for the voltage comparators at 2/3VCC for comparator A and 1/3VCC for comparator B. When the threshold input is above 1/3VCC, the flip-flop is reset, and when the trigger input is above 2/3VCC, the flip-flop is set. The ouptut comes from an interface/driver circuit which is driven by the flip-flop. The output

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is capable of sourcing or sinking current. The discharge output is used to discharge external timing capacitors as we will see when we discuss specific applications.

Figure 6. Functional block diagram of the 555 timer circuit While the 555 timer is touted as being capable of sinking or sourcing 200 mA, that is with a 15 volts power supply and at that, the output voltage rises or falls substantially (typically 2.5 volts) from the rails. With a power supply of 5 volts, the consumer version of the chip can sink only 5 mA; still capable of driving TTL loads. Consult the data sheets from several manufacturers for complete specifications. Monostable Application In the this application, with the external connections shown in Figure 7, the timer circuit will operate as a single-shot multivibrator. Here, if the trigger gets a negative pulse, the flip-flop is set, making Q’ high, truning off the discharge transistor, which then allows the capacitor to be charged up toward VCC. When the capacitor voltage reaches 2/3VCC, the threshold signal causes the flip-flop to be reset, discharging the capacitor again. Typical waveforms are shown in Figure 8. It can be seen from the waveforms that the ouput remains low until a trigger signal is received. Then the output goes high while the capacitor charges and then goes back low where it remains until another trigger pulse is received. Hence, the name single-shot. Multiple triggers or continuous low voltage on the trigger input during charging have no effect, but the trigger signal must go back high again before the flip-flop can be reset by the threshold signal. Timing is dependent on the time it takes the capacitor to charge up from a discharged state, or very near zero volts, to 2/3VCC. The charging equation is vC(t) = VCC(1 - e-t/RC) We solve this equation for VC = 2/3VCC. 0.667VCC = VCC(1 - e-t/RC) and solve for t

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1 - 0.667 = e-t/RC or t/RC = - ln(0.667) = 1.099 or the length of the pulse is t = 1.099RC

Figure 7. Connections for monostable operation.

Figure 8. Wveforms for monostable operation. The capacitor is shown as being discharged intstantaneously. However, the discharge current, and hence the rate, is limited by the β of the discharge transistor. In this mode of operation, the discharge is accomplished quickly, but it does limit how soon the cycle can be restarted. If a trigger signal is received during this discharge time, the flip-flop will be set and the capacitor recharged, this time, starting before it is completely discharged and shortening the output pulse.

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Astable Application Figure 9 shows the connections for astable or free-running operation. Both the trigger and threshold inputs are connected directly to the capacitor. There is an additional resistor, RB, connected between the capacitor and the discharge transistor to slow the discharge. When the capacitor discharges to 1/3VCC, the trigger comparator switches and sets the flip-flop which in turn turns off the discharge transistor, allowing the capacitor to start charging up through both resistors, RA and RB. When the capacitor reaches 2/3VCC, the threshold input causes the flip-flop to reset which in turn turns on the discharge transistor and the capacitor discharges again. Thus, the capacitor charges and discharges back and forth between 1/3VCC and 2/3VCC. Typical waveforms are shown in Figure 10.

Figure 9. Astable Circuit

Figure 10. Astable waveforms The discharge equation, starting at 2/3VCC is t

− 2 v C (t ) = VCC e RB C 3 The capacitor discharges to 1/3VCC at time = tD, the time during discharge

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tD

− 1 2 VCC = VCC e RB C 3 3

tD = RBC(-ln0.5) = 0.693RBC During charging, the capacitor starts at 1/3VCC, and charges toward VCC. The equation is t

v C (t ) = VCC

− 2 − VCC e ( RB + R A ) C 3

The capacitor charges up to 2/3VCC at time = tC, the time during charge tC

− 2 2 VCC = VCC − VCC e ( RB + R A ) C 3 3

tC = C(RA + RB)(-ln0.5) = 0.693(RA + RB)C The total period is the sum of the charge and discharge time tT = 0.693(RA + 2RB)C It is important to note that the charging time will always be laronger than the discharging time with the result that the timer output will always be high longer than it will be low. From the equations, the only way to make a square wave is to make RA = 0. But since RA is connected from VCC to the collector of the discharge transistor, that would result in very large currents through that transistor, which cannot be allowed. Also note that the time is independent of the supply voltage, a very desirable result. The accuracy of the time, however, is dependent on not only the accuracy of the external resistor and capacitor values, but is dependent on the accuracy of the internal voltage divider string that set the voltage comparator reference levels for charge and discharge trigger points. This 3 resistor voltage divider string is made on the chip simultaneously so that the values tend to track. Any deviation from nominal value of one resistor tends to be seen by all resistors in the string so that the reference voltages are fairly accurate. The control voltage input is connected directly to the reference voltage for the threshold voltage comparator. By external manipulation of this voltage, the thresholds for charge and discharge can be manipulated. The trigger point for discharge will be one-half the charge trigger point. Does this control give the designer a means by which to make the charge and discharge times equal, producing a square wave at the timer output?

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Exercises

1. You have a 0.1 µF, ±10% capacitor, all standard value 5% resistors, and a 555 timer integrated circuit. Design a monostable multivibrater circuit with a nominal pulse width of 20 ms. What are the worst case minimum and maximum pulse widths? 2. With the same components from the previous exercise, design an astable timing circuit with a nominal frequency of 1kHz.

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APPENDIX A DEFINITION OF TERMS FOR LOGIC GATE SPECIFICATIONS VinLmax

The maximum input voltage that will be seen as a "LOW".

VinHmin

The minimum input voltage that will be seen as a "HIGH".

VoL Output voltage when the output is "LOW". Spec's often give a VoLmax which is the maximum value you will normally find, considering component variations. VoH

Output voltage when the output is "HIGH".

Note that the output voltage specifications are given under some specific current condition, usually maximum current. The VoH term is almost always associated with IoH. IinL The current entering the input terminal when the input voltage is low. Normally specified at a fixed input voltage (Usually at VoL ). This current is usually negative; leaving the input terminal. IinH The current entering the input terminal when the input voltage is high. In the cases of DTL and TTL circuits, the input current is specified at maximum input voltage which results in maximum input circuit. IoH The current entering the output terminal when the output voltage is high. Note that this current is usually negative, the current actually leaves the output terminal. This current is usually specified when the output voltage is at its minimum, high-level value, VoHmin. What this definition means is that this is the maximum current you can take out of the output and still have the output voltage stay at or above the minimum specified value. IoL The current that the gate can sink when the output is low. Since the gate usually has a transistor driver at the output, the low-level output occurs when the transistor is saturated. Thus, this current is specified at the largest current that the transistor can sink and still stay in saturation. As discussed in the book, while the transistor is in saturation, the voltage rises as the current increases. Thus, this current is usually specified at some specific VoLmax, the maximum output voltage when the output is low (and sinking this large current). Current polarity is defined as positive going into a terminal. Voltages are measured from the terminal to ground. The subscript refer the terminal and its logic level. For example, IoL is the output current when the output is low. and IinL is the input current when the input is low. Figure A1. Terminal definitions

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APPENDIX B NOISE MARGINS NML Noise Margin Low. NML = VinLmax - VoLmax. This is the amount of noise voltage that can appear on the low level output signal of a gate and still have that low level signal be guaranteed to be recognized as a low at the input of a load gate. NMH Noise Margin High. NMH = VoHmin - VinHmin. This is the amount of noise voltage that can appear on a high level signal at the output of a gate and still have that high level signal be guaranteed to be recognized as a high at the input of a load gate.

Figure B1. Definition of noise margins

Figure B2. Typical voltage transfer characteristic for an inverting gate

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APPENDIX C PROPAGATION DELAYS Propagation delays are measured from the 50% points of the voltage transitions bwtween the input and output waveforms. The example waveforms given below are for an inverting gate, but measuremnts are taken the same way for non-inverting gattes. The subscripts HL and LH refer to the transitions of the output waveform, low-high and highlow respectively.

Figure C1. Propagation delay measurements

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APPENDIX D DESIGN EXERCISES D1. Integrated Circuit Resistor Mountaineer Electronics is developing a line of logic integrated circuits and your job is to design a 5000 Ohm resistor to be made simultaneously with the IC transistors. The resistor will be made by diffusing donors into a p-type substrate. Connections to the resistor will be made by attaching conducting material to the ends of the resistor with heavily doped ohmic contacts.

The length of the resistor is L and the cross-sectional area is WxDepth. Due to process limitations, the minimum dimension on the surface, W and L, is 0.01 mm. Several process are available with 0.25, 0.5, 1.0, 2.0, and 4.0 x 1015 donors/cm3. For each of these processes, the time required in the high temperature oven is proportional to the doping level. For example, for the lowest doping level, 0.25x1015, the oven time is 30 minutes, and for 0.5x1015 doping level, the oven time is 60 minutes. Diffusion depth is proportional to the time in the oven at 0.01mm/hr. Cost is a combination of • silicon surface area at $2/cm2 • oven time at $1/hr-cm2 Design a 5000 Ohm resistor for minimum cost. The format required for submission of design exercises is given at the end of the listing of design exercises. D2. LED POWER INDICATOR McMee Toys Inc. is developing a talking teddy bear toy, the “Hippy-Dippy Teddy Bear”. The toy is to use two ME56 LEDs for the eyes that light up when the toy is active which are turned on by a simple switch and are powered from four 1.5 volt batteries, for a total of 6V. Testing on the LEDs in the lab indicate that the current must be kept above 15 mA to be bright enough. You must design a circuit that will keep the current above 15 mA but also must not exceed the ratings given in the specifications.

Appendix

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Design for minimum cost while guaranteeing operation within specifications over +/- 5% resistor tolerance and specified range of electrical characteristics of the LEDs. This type of design is called worst-case design. It is intended to be used to guarantee the circuit will operate over all possible values of components. To do worst-case design, you must select the extreme possible values of all components that will present the worst cases. In this case, you must use the minimum LED forward voltage and minimum resistance values to determine maximum current and vice-versa. The design engineer has substantial control over the quality of the finished device by how well it is designed. For example, if only typical or nominal values are used in the design calculations, then as components are selected from the bin for the manufacturing operation, many combinations of components would cause the circuit to operate outside its specifications and would likely fail prematurely if it operated at all. Rework is extremely expensive and is often much more expensive than the selling price of the product. The implications for failure of marketed devices is substantial. One dissatisfied customer has much greater effect on the profitability of the business than the purchase price of the original product. The implications go even further than company profits in that profits are necessary to keep employees working and even have effects on the economic health of the community. D3. SOLENOID DRIVER The Hippy-Dippy Teddy Bear from McMee Toys will wiggle its ears when the child rubs its tummy. Actually, a sensitive switch in the bear’s tummy is activated with a slight pressure on its tummy. When the switch is depressed, a solenoid pulls the ears backward slightly, and when the switch is released, the solenoid is deactivated and a spring pulls the ears back to the normal position. The effect is (supposed to be anyway) that the ears wiggle when the tummy is rubbed. Your job is to design the circuit to activate the solenoid when the switch button is depressed. Again, you must use worst-case design using the components from the list in the appendix. D4. Logic Interface The Hippy-Dippy Teddy Bear from McMee Toys will alternately blink its eyes (LEDs) when its left hand is squeezed. The left hand has a sensitive switch installed which is depressed when the hand is squeezed. Internally, there is a 1 Hz clock signal which runs the logic to make the appropriate logic signals for blinking the lights. Assume a 6 volt (from batteries) and a 5-volt power supply are available. The ME56 LEDs must have a minimum of 20 mA when on. Operating temperature range is 0-70 0C. Design the system using 74LS00 and 74LS04 gates which take the 1 Hz signal and the switch input to alternately blink the LEDs. When the switch is not depressed, both LEDs

Appendix

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must be on continuously. The switch interface circuit must provide two logic levels in its two states. You may use either high or low logic when the switch is depressed, but you must specify which and you must specify how to wire the switch. The 74LS00 and 74LS04 ICs cost $1.25 each, including insertion and PCB space. D5. RMS VOLTMETER D’Arsonval meters move only in response to average voltage. Since the average of a sine wave is zero, these meters cannot directly measure the RMS value of a sine wave. Instead, a diode bridge is used to full-wave rectify the sine wave which will have a nonzero average value. Then resistors are used to scale the meter to provide a reading from the average of the full-wave rectified signal to the appropriate RMS value. Such a meter provides accurate scaling only for sine waves. Fifty-Six Electronics Company is designing an old-fashioned multimeter using a 50 millivolt full-scale D’Arsonval meter with a 5000 Ohm coil. Your job is to design the meter to have three scales: 2, 20, and 200 volts RMS full-scale deflection for sine waves. You may use a diode model IF = 0 for VF 0. In addition, you may use standard 1% resistors. You have a rotary switch to select scales.

Diode Bridge Circuit

Rotary Switch

Meter Equivalent Circuit

D6. Transducer - Logic Interface The Hippy-Dippy Teddy Bear from McMee Toys, Inc., responds to temperatures around 70 0F. When the temperature drops below 70, a voice circuit asks for a blanket. When the temperature goes back up, the voice circuit responds by saying it is too warm. The temperature transducer provides a 1.00 volt output signal when the temperature is above 70 degrees and zero volts when below 70 degrees. The transducer Thevenin equivalent resistance is 1000 Ohms. Design an interface that will read the transducer outputs and provide appropriate logic levels to drive two 7400 gates. Note that when the transducer output is low, the gates must see a valid logic level, and when the transducer output is high, the gates must see the other logic level.

Appendix

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D7. MOSFET INVERTER Mountaineer Electronics is developing an integrated circuit using NMOS logic with a 3 volt power supply. To accomodate TTL loads, the output inverter stage will use an integrated circuit resistor pull-up (to 3 volts) and an n-channel MOSFET. Design the output inverter devices, the resistor and n-channel MOSFET to drive two 74LS00 gates. Noise margins must be at least 0.5 volts. The process for MOSFETs gives a k’ = 0.025 mA/V2 and VT = 1.0 Volts. You must determine the W and L for the channel of the MOSFET. The process parameters for the resistor are given in problem D1. In both devices, the minimum feature size (W and L) is 3.0 microns and can only be made in increments of 1 micron. D8. CMOS Nand Gate Instead of the resistor inverter in problem D7, design a 2-input Nand gate output stage to drive four 74LS00 loads with a noise margin of at least 0.4 volts. The CMOS power supply is 3 volts and the process provides VT = 1.0 volt, and k’ = 0.040 mA/V2. The minimum feature size is 2.0 microns with increments of 1.0 micron. D9. DTL Nand Gate Mountaineer Electronics is developing a DTL logic family for use in automotive applications. The automotive environment requires the system to operate in a temperature range of -40 0C to +150 0C and a power supply voltage of 9-15 volts. Design a DTL Nand gate to operate in this environment. Because the ignition system on the automobile creates a lot of noise, the noise margins must be at least 2 volts.

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DESIGN EXERCISE FORMAT The following is the format required when submitting design exercises. Please follow it. It has been developed to minimize the amount of writing for the student and reading for the grader.

EE 56 D1, Solid State Resistor Design

Date________, Name_________

DESIGN OBJECTIVE: Design a solid state resistor, n-type material on a p-type substrate. Performance Requirements: R = 5 KΩ Design Constraints: Doping Levels ND = 0.25, 0.5, 1.0, 2.0, and 4.0 x 1016 donors/cm3 Silicon Area Charge $2/cm2 Oven Time Charge $500/hr/m2 , t = kND2 Diffusion Depth 0.01 mm/hr Minimum surface dimension 0.01 mm Design Criteria: Minimum Cost DESIGN COMPOSITION Make a drawing here showing dimensions Provide all specifications so a technician can fabricate the device W = _____, L = _______, Doping = _______, (Oven time = _____), (Diffusion Depth = ___) Predicted Performance (Exactly how does your device perform?) Resistance = ____________ Cost = ________ PROOF OF PERFORMANCE Resistance Calculation Cost Calculation Proof of peformance is not a description of how you arrived at the component values, but is a proof that your values will cause the device to perform as you predict. The proof may be calculation, simulation etc.. In most cases you will have to show the models used for devices or other assumptions made.

Appendix

8

Copper Wire Tables Standard Annealed Copper, American Wire Gauge 20 0C

Gauge AWG

Diameter (mm)

Ohms per Kilometer

Kg per Kilometer

0000 000 00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 37 39 40

11.68 10.40 9.266 8.252 7.348 6.543 5.827 5.189 4.620 4.115 3.665 3.264 2.906 2.588 2.30 2.05 1.83 1.63 1.45 1.29 1.15 1.02 0.912 0.813 .724 .643 .574 .511 .455 .404 .361 .320 .287 .254 .226 .203 .180 .160 .142 .127 .114 .102 .089 .079

0.1608 0.2028 0.2557 0.3223 0.4065 0.5128 0.6466 0.8152 1.028 1.297 1.634 2.061 2.600 3.277 4.14 5.21 6.56 8.28 10.4 13.2 16.6 21.0 26.4 33.2 41.9 53.2 66.6 84.2 106 135 169 214 266 340 430 532 675 857 1090 1360 1680 2130 2780 3540

953.2 755.8 599.5 475.5 377.0 298.9 237.1 188.0 149.0 118.2 93.80 74.38 58.95 46.77 37.1 29.4 23.4 18.5 14.7 11.6 9.24 7.32 5.81 4.61 3.66 2.88 2.30 1.82 1.44 1.14 0.908 .715 .575 .450 .357 .288 .227 .179 .141 .113 .0912 .0721 .0552 .0433

Appendix

9

RESISTORS 5% Standard Sequence 10 16 27 43 11 18 30 47 12 20 33 51 13 22 36 56 15 24 39 62

Cost = $0.25 Including Insertion and PCB space 68 75 82 91 Power ratings: 1/8, 1/4, 1/2, 1, and 2 watts

Obtain standard values by multiplying by a power of 10 (10-1, 100, 101, 102, etc.) Values available: 1 ≤ R ≤ 10 M Ohms Temperature Coefficient: -0.025 ≤ %/0C ≤ +0.075

1% Standard Sequence Cost = $0.70 including insertion and PCB space 10.0 15.4 23.7 36.5 56.2 86.6 10.2 15.8 24.3 37.4 57.6 88.7 10.5 16.2 24.9 38.3 59.0 90.9 10.7 16.5 25.5 39.2 60.4 93.1 11.0 16.9 26.1 40.2 61.9 95.3 11.3 17.4 26.7 41.2 63.4 97.6 11.5 17.8 27.4 42.2 64.9 11.8 18.2 28.0 43.2 66.5 12.1 18.7 28.7 44.2 68.1 12.4 19.1 29.4 45.3 69.8 12.7 19.6 30.1 46.4 71.5 13.0 20.0 30.9 47.5 73.2 13.3 20.5 31.6 48.7 75.0 13.7 21.0 32.4 49.9 76.8 14.0 21.5 33.2 51.1 78.7 14.3 22.1 34.0 52.3 80.6 14.7 22.6 34.8 53.6 82.5 15.0 23.2 35.7 54.9 84.5 Obtain standard values by multiplying by a power of 10 (10-1, 100, 101, 102, etc.) Available Values: 10 ≤ R ≤ 10 M Ohms Temperature Coefficient: ± 100 ppm/0C

Appendix

10

ME56 LED Specifications Visible Red LED

Cost $1.75

Maximum Ratings: Continuous Forward Current, IF Reverse Voltage, VR Operating Temperature Range

Mountaineer Electronics Inc. Released 12/27/00

35 mA 2.5 Volts -40 - +100 0C

Electrical Characteristics Characteristic Min Reverse Leakage Current (VR = 2.5V) Forward Voltage (IF = 35 mA),VF 0.8 Temperature Coefficient, Forward Voltage Capacitance (f = 1 MHz)

Typ 100 1.8 -2.2 50

Max 2.2

Unit nA V mV/0K pF

ME57 Silicon Diode Specifications General Purpose Diode Cost $0.75

Mountaineer Electronics Inc. Released 12/27/00

Maximum Ratings: Continuous Forward Current Reverse Voltage Operating Temperature Range

100 mA 100 Volts -40 - +150 0C

Electrical Characteristics Characteristic Reverse Leakage Current @VR = 100V Forward Voltage (IF = 100 mA) Temperature Coefficient, forward voltage Capacitance (f = 1 MHz)

Appendix

Min 0.50

Typ 1 0.60 -2.0 25

Max 100 0.75

Unit nA V mV/0K pF

11

ME58 Silicon Zener Diode Specifications Zener Diode Cost $1.20 Maximum Ratings: Continuous Forward Current, IF Continuous Zener Current, IZ Maximum Power Dissipation Operating Temperature Range Electrical Characteristics (@25 0C) Characteristic Forward Voltage (IF = 50 mA) Zener Voltage (IZ = 10 mA), VZ Temperature Coefficient, Zener Voltage Capacitance (f = 1 MHz)

Appendix

Mountaineer Electronics Inc. Released 12/30/00

50 mA 10 mA 25 mW -40 - +150 0C

Min 0.40 2.05

Typ 2.20 -2.0 50

Max 1.00 2.35

Unit V V mV/0K pF

12

ME256 NPN BJT Specifications NPN General Purpose Transistor

Cost $1.50

Maximum Ratings: Collector-Emitter Voltage Collector-Base Voltage Emitter-Base Voltage Continuous Collector Current Total Device Dissipation @TA = 25 0C Derate Above 25 0C Operating Temperature Range (TJ)

Mountaineer Electronics Inc. Released 12/29/00

40 V 60 V 5.0 V 200 mA 625 mW 5 mW/0C -40 - +150 0C

Thermal Characteristics Max 83.3 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 200 0C/W Electrical Switching Characteristics Off Characteristics Collector-Emitter Breakdown Voltage (IC= 1.0 mA, IB = 0) Collector-Base Breakdown Voltage (IC = 10µA, IE = 0)

Min 40

Typ

Max

60

Unit V

V

Base Cutoff Current (VCE = 30 V, VEB = 3 V)

1

50

nA

Collector Cutoff Current (VCE = 30 V, VEB = 3 V)

1

50

nA

Base-Emitter Cut-In Voltage, VBEγ (VCE = 30 V, IC = 0.01 mA)

0.45

0.50

Saturation Characteristics Collector-Emitter Saturation Voltage, VCEsat IC = 1.0 mA, IB = 0.10 mA IC = 50.0 mA, IB = 5.0 mA Base-Emitter Saturation Voltage, VBEsat IC = 1.0 mA, IB = 0.10 mA IC = 50 mA, IB = 5.0 mA

Appendix

0.55

0.65 0.75

V

0.20 0.30

V V

0.95

V V

13

ME257 PNP BJT Specifications PNP General Purpose Transistor

Cost $1.60

Maximum Ratings: Collector-Emitter Voltage Collector-Base Voltage Emitter-Base Voltage Continuous Collector Current Total Device Dissipation @TA = 25 0C Derate Above 25 0C Operating Temperature Range (TJ)

Mountaineer Electronics Inc. Released 12/29/00

-40 V -60 V -5.0 V -200 mA 625 mW 5 mW/0C -40 - +150 0C

Thermal Characteristics Max 83.3 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 200 0C/W Electrical Switching Characteristics Off Characteristics Collector-Emitter Breakdown Voltage (IC= -1.0 mA, IB = 0) Collector-Base Breakdown Voltage (IC = -10µA, IE = 0)

Min -40

Typ

Max

-60

Unit V

V

Base Cutoff Current (VCE = -30 V, VEB = -3 V)

-1

-50

nA

Collector Cutoff Current (VCE = -30 V, VEB = -3 V)

-1

-50

nA

Base-Emitter Cut-In Voltage, VBEγ (VCE = -30 V, IC = -0.01 mA) Saturation Characteristics Collector-Emitter Saturation Voltage, VCEsat IC = -1.0 mA, IB = -0.10 mA IC = -50.0 mA, IB = -5.0 mA Base-Emitter Saturation Voltage, VBEsat IC = -1.0 mA, IB = -0.10 mA IC = -50 mA, IB = -5.0 mA

Appendix

-0.45 -0.55

V

-0.20 V -0.35 V

-0.55 -0.65 V -0.75 -0.95 V

14

ME181 N-Channel Enhancement Mode MOSFET Specifications Mountaineer Electronics Inc. General Purpose NMOS Transistor Cost $0.75 Released 12/29/01 Maximum Ratings: Drain-Source Voltage Gate-Source Voltage Continuous Drain Current Total Device Dissipation @TA = 70 0C Derate Above 70 0C Operating Temperature Range (TJ)

50 V +/- 20 V 200 mA 240 mW 3 mW/0C -65 - +150 0C

Thermal Characteristics Max 40 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 333 0C/W Electrical Switching Characteristics Off Characteristics Drain-Source Breakdown Voltage, (VGS = 0V, ID = 50µA)

Min 50

Typ

Zero Gate Voltage Drain Current, IDDS (VDS = 50V, VGS = 0V) Gate-Source Leakage Current , IGSS (VGS = 20 V, VDS = 0 V) On Characteristics Gate Threshold Voltage, VGS(th) VDS = 10 V, ID = 5 µA

1.0

Gate Threshold Temperature Coefficient

0.3

Gate Quiescent Voltage, VGS(q) VDS = 28 V, ID = 2.00 mA

3.5

Static Drain-Source On Resistance, RDSon VGS = 10V, ID = 100 mA

Appendix

Unit V

1

µA

1

µA

1.5

2.0

5.0

0.66

0.8

V

5.5

V

7.0

50

V mV/oK

-4

Drain-Source On Voltage, VDS(on) VGS = 10 V, ID = 50 mA

Forward Transconductance, gm VGS = 10V, VDS = 5V

Max

mhos

80

Ω

15

Dynamic Characteristics VDS = 28Vdc, VGS = 0V, f=1.0MHz Input Capacitance, Ciss Output Capacitance, Coss Reverse Transfer Capacitance, Crss

Appendix

13 7

pF pF

0.7

pF

16

ME182 P-Channel Enhancement Mode MOSFET Specifications Mountaineer Electronics Inc. General Purpose PMOS Transistor Cost $0.75 Released 12/29/01 Maximum Ratings: Drain-Source Voltage Gate-Source Voltage Continuous Drain Current Total Device Dissipation @TA = 70 0C Derate Above 70 0C Operating Temperature Range (TJ)

-50 V +/- 20 V -100 mA 240 mW 3 mW/0C -65 - +150 0C

Thermal Characteristics Max 40 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 333 0C/W Electrical Switching Characteristics Off Characteristics Drain-Source Breakdown Voltage, (VGS = 0V, ID = - 50µA)

Min -50

Typ

Zero Gate Voltage Drain Current, IDDS (VDS = -50V, VGS = 0V) Gate-Source Leakage Current , IGSS (VGS = -20 V, VDS = 0 V) On Characteristics Gate Threshold Voltage, VGS(th) VDS = -10 V, ID = -5 µA

-1.0

Gate Threshold Temperature Coefficient

Max

Unit V

-1

µA

-1

µA

-1.5

-2.0

V

3.2

mV/oK

Drain-Source On Voltage, VDS(on) VGS = -10 V, ID = -50 mA

-0.3

-0.66 -0.8

V

Gate Quiescent Voltage, VGS(q) VDS = -28 V, ID = -200.0 mA

-3.5

-5.5

V

Forward Transconductance, gm VGS = -10V, VDS = -5V Static Drain-Source On Resistance, RDSon VGS = -10V, ID = -100 mA

Appendix

5.0

7.0

50

mhos

80

Ω

17

Dynamic Characteristics VDS = -28Vdc, VGS = 0V, f=1.0MHz Input Capacitance, Ciss Output Capacitance, Coss Reverse Transfer Capacitance, Crss

Appendix

14 8

pF pF

0.7

pF

18

ME183 N-Channel Enhancement Mode MOSFET Specifications Mountaineer Electronics Inc. General Purpose NMOS Transistor Cost $1.75 Released 12/29/00 Maximum Ratings: Drain-Source Voltage Gate-Source Voltage Continuous Drain Current Total Device Dissipation @TA = 70 0C Derate Above 70 0C Operating Temperature Range (TJ)

65 V +/- 20 V 2A 35 W 275 mW/0C -65 - +150 0C

Thermal Characteristics Max 5.4 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 40 0C/W Electrical Switching Characteristics Off Characteristics Drain-Source Breakdown Voltage, (VGS = 0V, ID = 50µA)

Max

Unit V

Zero Gate Voltage Drain Current, IDDS (VDS = 28V, VGS = 0V)

1

mA

Gate-Source Leakage Current , IGSS (VGS = 20 V, VDS = 0 V)

1

mA

4.0

V

On Characteristics Gate Threshold Voltage, VGS(th) VDS = 10 V, ID = 50 µA

Min 65

2.0

Gate Threshold Temperature Coefficient

0.3

Gate Quiescent Voltage, VGS(q) VDS = 28 V, ID = 200.0 mA

3.5

Static Drain-Source On Resistance, RDSon VGS = 10V, ID = 2A

Appendix

3.6

mV/oK

-4

Drain-Source On Voltage, VDS(on) VGS = 10 V, ID = 0.50 A

Forward Transconductance, gm VGS = 10V, VDS = 5V

Typ

5.0

0.66

0.8

V

5.5

V

7.0

1.5

mhos

2.0

Ω

19

Dynamic Characteristics VDS = 28Vdc, VGS = 0V, f=1.0MHz Input Capacitance, Ciss Output Capacitance, Coss Reverse Transfer Capacitance, Crss

Appendix

13 7

pF pF

0.7

pF

20

ME184 P-Channel Enhancement Mode MOSFET Specifications Mountaineer Electronics Inc. General Purpose PMOS Transistor Cost $1.75 Released 12/29/00 Maximum Ratings: Drain-Source Voltage Gate-Source Voltage Continuous Drain Current Total Device Dissipation @TA = 70 0C Derate Above 70 0C Operating Temperature Range (TJ)

-65 V +/- 20 V -2 A 40 W 300 mW/0C -65 - +150 0C

Thermal Characteristics Max 7.4 0C/W Thermal Resistance, Junction-Case RθJC Thermal Resistance, Junction-Ambient RθJA 45 0C/W Electrical Switching Characteristics Off Characteristics Drain-Source Breakdown Voltage, (VGS = 0V, ID = - 50µA)

Min -65

Typ

Zero Gate Voltage Drain Current, IDDS (VDS = -28V, VGS = 0V) Gate-Source Leakage Current , IGSS (VGS = -20 V, VDS = 0 V) On Characteristics Gate Threshold Voltage, VGS(th) VDS = -10 V, ID = -50 µA

-2.0

Gate Threshold Temperature Coefficient

Max

Unit V

-1

mA

-1

mA

-3.4

-4.0

V

3.2

mV/oK

Drain-Source On Voltage, VDS(on) VGS = -10 V, ID = -0.50 A

-0.3

-0.66 -0.8

V

Gate Quiescent Voltage, VGS(q) VDS = -28 V, ID = -200.0 mA

-3.5

-5.5

V

Forward Transconductance, gm VGS = -10V, VDS = -5V Static Drain-Source On Resistance, RDSon VGS = -10V, ID = -2A

Appendix

5.0

7.0

1.75

mhos

2.5

Ω

21

Dynamic Characteristics VDS = -28Vdc, VGS = 0V, f=1.0MHz Input Capacitance, Ciss Output Capacitance, Coss Reverse Transfer Capacitance, Crss

Appendix

14 8

pF pF

0.7

pF

22

SOL-1 Electro-Mechanical Solenoid Cost $2.65

Mountaineer Electronics Released 12/29/00

Coil Resistance Inductance 1 Inch Extension 1/2 Inch Extension Closed

850 +/- 10% Ohms

Pull-in Characteristics: 1 Inch Extension Pull-in force @12 mA @ 30 mA

Min 8.0 30

Oz Oz

1/2 Inch Extension Pull-in Force @ 12 mA @ 30 mA

30 75

Oz Oz

Holding Force@ 12 mA @ 30 mA

120 300

Oz Oz

Maximum Release Current@ 4 Oz

1.5 mA

50 120 200

mH mH mH

Maximum Continuous Current 20 Pulse Mode Current (2 second max) 40 (< 10% duty cycle) SW-1 Sensitive Switch

Cost $2.25

Electrical Characteristics Maximum Voltage (non-inductive) Maximum Current Mechanical Characteristics Button Travel Maximum Force to Fully Depress Button SPDT DPDT

mA mA

Mountaineer Electronics Released 12/29/00

25 V 6.0 mA

1.0 mm +/- 0.1 mm 2.0 Oz 2.5 Oz

Electrical diagram for SPDT

Appendix

23